RD Sharma Solutions For Class 12 Maths Exercise 5.3 Chapter 5 Algebra of Matrices

RD Sharma Class 12 Solutions Chapter 5 Ex 5.3 PDF Free Download

RD Sharma Solutions for Class 12 are available at BYJU’S website in PDF format with solutions prepared by experienced faculty in a precise manner. It can mainly be used by the students as a study material to clear the Class 12 exams with a good score according to the CBSE syllabus. Exercise 5.3 of the fifth chapter contains problems, which are solved based on the transpose of a matrix. The solutions prepared are in an explanatory manner to bring about conceptual clarity among the students. RD Sharma Solutions for Class 12 Maths Chapter 5 Algebra of Matrices Exercise 5.3 are provided here.

Download the PDF of RD Sharma Solutions For Class 12 Chapter 5 – Algebra of matrices Exercise 5.3

 

rd sharma class 12 maths chp 5 ex 3
rd sharma class 12 maths chp 5 ex 3 1
rd sharma class 12 maths chp 5 ex 3 2
rd sharma class 12 maths chp 5 ex 3 3
rd sharma class 12 maths chp 5 ex 3 4
rd sharma class 12 maths chp 5 ex 3 5
rd sharma class 12 maths chp 5 ex 3 6
rd sharma class 12 maths chp 5 ex 3 7
rd sharma class 12 maths chp 5 ex 3 8
rd sharma class 12 maths chp 5 ex 3 9
rd sharma class 12 maths chp 5 ex 3 10
rd sharma class 12 maths chp 5 ex 3 11
rd sharma class 12 maths chp 5 ex 3 12
rd sharma class 12 maths chp 5 ex 3 13
rd sharma class 12 maths chp 5 ex 3 14
rd sharma class 12 maths chp 5 ex 3 15
rd sharma class 12 maths chp 5 ex 3 16
rd sharma class 12 maths chp 5 ex 3 17
rd sharma class 12 maths chp 5 ex 3 18
rd sharma class 12 maths chp 5 ex 3 19
rd sharma class 12 maths chp 5 ex 3 20
rd sharma class 12 maths chp 5 ex 3 21
rd sharma class 12 maths chp 5 ex 3 22
rd sharma class 12 maths chp 5 ex 3 23
rd sharma class 12 maths chp 5 ex 3 24
rd sharma class 12 maths chp 5 ex 3 25
rd sharma class 12 maths chp 5 ex 3 26
rd sharma class 12 maths chp 5 ex 3 27
rd sharma class 12 maths chp 5 ex 3 28
rd sharma class 12 maths chp 5 ex 3 29
rd sharma class 12 maths chp 5 ex 3 30
rd sharma class 12 maths chp 5 ex 3 31
rd sharma class 12 maths chp 5 ex 3 32
rd sharma class 12 maths chp 5 ex 3 33
rd sharma class 12 maths chp 5 ex 3 34
rd sharma class 12 maths chp 5 ex 3 35
rd sharma class 12 maths chp 5 ex 3 36
rd sharma class 12 maths chp 5 ex 3 37
rd sharma class 12 maths chp 5 ex 3 38
rd sharma class 12 maths chp 5 ex 3 39
rd sharma class 12 maths chp 5 ex 3 40
rd sharma class 12 maths chp 5 ex 3 41
rd sharma class 12 maths chp 5 ex 3 42
rd sharma class 12 maths chp 5 ex 3 43
rd sharma class 12 maths chp 5 ex 3 44
rd sharma class 12 maths chp 5 ex 3 45
rd sharma class 12 maths chp 5 ex 3 46
rd sharma class 12 maths chp 5 ex 3 47
rd sharma class 12 maths chp 5 ex 3 48

 

Access other exercises of RD Sharma Solutions For Class 12 Chapter 5 – Algebra of Matrices

Exercise 5.1 Solutions

Exercise 5.2 Solutions

Exercise 5.4 Solutions

Exercise 5.5 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 5 – Algebra of Matrices Exercise 5.3

1. Compute the indicated products:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 203

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 204

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 205

Solution:

(i) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 206

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 207

(ii) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 208

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 209

(iii) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 210

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 211

2. Show that AB ≠ BA in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 212

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 213

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 214

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 215

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 216

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 217

From equation (1) and (2), it is clear that

AB ≠ BA

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 218

Now again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 219

From equation (1) and (2), it is clear that

AB ≠ BA

(iii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 220

Now again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 221

From equation (1) and (2), it is clear that

AB ≠ BA

3. Compute the products AB and BA whichever exists in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 222

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 223

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 224

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 225

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 226

BA does not exist

Because the number of columns in B is greater than the rows in A

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 227

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 228

(iii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 229

AB = [0 + (-1) + 6 + 6]

AB = 11

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 230

(iv) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 231

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 232

4. Show that AB ≠ BA in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 233

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 234

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 235

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 236

From equation (1) and (2), it is clear that

AB ≠ BA

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 237

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 238

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 239

From equation (1) and (2) it is clear that,

AB ≠ BA

5. Evaluate the following:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 240

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 241

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 242

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 243

First we have to add first two matrix,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 244

On simplifying, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 245

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 246

First we have to multiply first two given matrix,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 247

= 82

(iii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 248

First we have subtract the matrix which is inside the bracket,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 249

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 250

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 251

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 252

We know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 253

Again we know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 254

Now, consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 255

We have,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 256

Now, from equation (1), (2), (3) and (4), it is clear that A2 = B2= C2= I2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 257

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 258

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 259

Now we have to find,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 260

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 261

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 262

Solution:

Given

C:\Users\tnluser\Downloads\CodeCogsEqn (74).gif

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 264

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 265

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 266

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 267

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 268

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 269

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 270

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 271

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 272

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 273

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 274

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 275

We know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 276

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 277

Again we have,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 278

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 279

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 280

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 281

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 282

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 283

From equation (1) and (2) AB = BA = 03×3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 284

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 285

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 286

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 287

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 288

From equation (1) and (2) AB = BA = 03×3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 289

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 290

Now consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 291

Therefore AB = A

Again consider, BA we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 292

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 293

Hence BA = B

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 294

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 295

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 296

Now again consider, B2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 297

Now by subtracting equation (2) from equation (1) we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 298

16. For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 299

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 300

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 301

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 302

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 303

From equation (1) and (2), it is clear that (AB) C = A (BC)

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 304

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 305

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 306

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 307

From equation (1) and (2), it is clear that (AB) C = A (BC)

17. For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 308

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 309

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 310

Consider LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 311

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 312

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 313

From equation (1) and (2), it is clear that A (B + C) = AB + AC

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 314

Consider the LHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 315

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 316

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 317

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 318

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 319

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 320

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 321

Now consider RHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 322

From the above equations LHS = RHS

Therefore, A (B – C) = AB – AC.

19. Compute the elements a43 and a22 of the matrix:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 323

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 324

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 325

From the above matrix, a43 = 8and a22 = 0

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 326

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 327

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 328

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 329

Now, consider the RHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 330

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 331

Therefore, A3 = p I + q A + rA2

Hence the proof.

21. If ω is a complex cube root of unity, show that

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 332

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 333

It is also given that ω is a complex cube root of unity,

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 334

We know that 1 + ω + ω2 = 0 and ω3 = 1

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 335

Now by simplifying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 336

Again by substituting 1 + ω + ω2 = 0 and ω3 = 1 in above matrix we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 337

Therefore LHS = RHS

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 338

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 339

Consider A2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 340

Therefore A2 = A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 341

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 342

Consider A2,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 343

Hence A2 = I3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 344

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 345

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 346

= [2x + 1 + 2 + x + 3] = 0

= [3x + 6] = 0

= 3x = -6

x = -6/3

x = -2

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 347

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 348

On comparing the above matrix we get,

x = 13

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 349

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 350

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 351

⇒ [(2x + 4) x + 4 (x + 2) – 1(2x + 4)] = 0

⇒ 2x2 + 4x + 4x + 8 – 2x – 4 = 0

⇒ 2x2 + 6x + 4 = 0

⇒ 2x2 + 2x + 4x + 4 = 0

⇒ 2x (x + 1) + 4 (x + 1) = 0

⇒ (x + 1) (2x + 4) = 0

⇒ x = -1 or x = -2

Hence, x = -1 or x = -2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 352

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 353

By multiplying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 354

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 355

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 356

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 357

Now we have to prove A2 – A + 2 I = 0

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 358

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 359

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 360

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 361

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 362

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 363

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 364

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 365

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 366

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 367

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 368

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 369

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 370

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 371

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 372

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 373

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 374

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 375

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 376

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 377

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 378

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 379

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 380

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 381

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 382

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 383

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 384

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 385

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 386

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 387

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 388

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 389

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 390

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 391

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 392

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 393

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 394

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 395

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 396

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 397

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 398

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 399

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 400

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 401

I is identity matrix, so

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 402

Also given, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 403

Now, we have to find A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 404

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 405

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 406

Now, we will find the matrix for 8A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 407

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 408

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 409

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 410

Substitute corresponding values from eqn (i) and (ii), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 411

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 412

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

Hence,

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 413

Therefore, the value of k is 7

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 414

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 415

To show that f (A) = 0

Substitute x = A in f(x), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 416

I is identity matrix, so 

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 417

Now, we will find the matrix for A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 418

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 419

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 420

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 421

Now, we will find the matrix for 2A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 422

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 423

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424

Substitute corresponding values from eqn (ii) and (iii) in eqn (i), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 426

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 427

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 428

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 429

Hence Proved

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 430

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 431

So 

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 432

Now, we will find the matrix for A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 433

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 434

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 435

Now, we will find the matrix for λ A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 436

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 437

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 438

But given, A2 = λ A + μ I

Substitute corresponding values from equation (i) and (ii), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 439

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 440

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

Hence, λ + 0 = 4 ⇒ λ = 4

And also, 2λ + μ = 7

Substituting the obtained value of λ in the above equation, we get

2(4) + μ = 7 ⇒ 8 + μ = 7 ⇒ μ = – 1

Therefore, the value of λ and μ are 4 and – 1 respectively

39. Find the value of x for which the matrix product

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 441

Solution:

We know,

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 442

is identity matrix of size 3.

So according to the given criteria

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 443

Now we will multiply the two matrices on LHS using the formula cij = ai1b1j + ai2b2j + … + ain bnj, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 444

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 445

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

So we get

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 446

So the value of x is 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 447


Leave a Comment

Your email address will not be published. Required fields are marked *