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Byju's Answer

Standard XII

Mathematics

Identity in Binary Operation

If A=[[ 2 5; ...

Question

If $A = [\begin{matrix} 2 & 5 1 & 3 \end{matrix}], B = [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$ and $I$ is Identity matrix of same order and $A^{t}$ is the transpose of matrix $A$ find $A^{t} . B + B I$ [4 Marks]

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Solution

$A = [\begin{matrix} 2 & 5 1 & 3 \end{matrix}]$
$∴ A^{t} = [\begin{matrix} 2 & 1 5 & 3 \end{matrix}]$ [1 Mark]
$A^{t} . B = [\begin{matrix} 2 & 1 5 & 3 \end{matrix}] [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$
$= [\begin{matrix} 2 \times 4 + 1 \times (- 1) & 2 \times (- 2) + 1 \times 3 3 \times 4 + 3 \times (- 1) & 5 \times (- 2) + 3 \times 3 \end{matrix}]$
$= [\begin{matrix} 7 & - 1 17 & - 1 \end{matrix}]$ [1 Mark]
$B . I = [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}] [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$
$= [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$ [1 Mark]
$∴ A^{t} . B + B I = [\begin{matrix} 7 & - 1 17 & - 1 \end{matrix}] + [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$
$= [\begin{matrix} 11 & - 3 16 & 2 \end{matrix}]$ [1 Mark]

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Similar questions

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} - 1 5 - 2 \end{matrix} \begin{matrix} 271 \end{matrix} \begin{matrix} 391 \end{matrix} ⎤ ⎥ ⎦

and

B = ⎡ ⎢ ⎣ \begin{matrix} - 4 11 \end{matrix} \begin{matrix} 123 \end{matrix} \begin{matrix} - 5 01 \end{matrix} ⎤ ⎥ ⎦

, then verify that

(i) {(A + B)}^{T} = A^{T} + B^{T}

(i i) {(A - B)}^{T} = A^{T} - B^{T}

Q. For the matrices

A

and

B

, verify that

{(A B)}^{'} = B^{'} A^{'}

where
(i)

A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦, B = [\begin{matrix} - 1 & 2 & 1 \end{matrix}]

(ii)

A = ⎡ ⎢ ⎣ \begin{matrix} 012 \end{matrix} ⎤ ⎥ ⎦, B = [\begin{matrix} 1 & 5 & 7 \end{matrix}]

Q. If

A = ⎡ ⎢ ⎣ \begin{matrix} 151 \end{matrix} \begin{matrix} 20 - 1 \end{matrix} \begin{matrix} - 3 21 \end{matrix} ⎤ ⎥ ⎦, B = ⎡ ⎢ ⎣ \begin{matrix} 342 \end{matrix} \begin{matrix} - 1 20 \end{matrix} \begin{matrix} 253 \end{matrix} ⎤ ⎥ ⎦

and

C = ⎡ ⎢ ⎣ \begin{matrix} 401 \end{matrix} \begin{matrix} 13 - 2 \end{matrix} \begin{matrix} 223 \end{matrix} ⎤ ⎥ ⎦

, then compute

(A + B)

and

(B - C)

Also, verify that

A + (B - C) = (A + B) - C

Q. (a)

If $H = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 2 & 3 & 1 3 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦, G = ⎡ ⎢ ⎣ \begin{matrix} 2 & 6 & - 10 0 & - 4 & 8 2 & - 8 & 4 \end{matrix} ⎤ ⎥ ⎦$

Find $3 H - \frac{1}{2} G .$
$[3]$

(b) If $3 A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 2 2 & 1 & - 2 x & 2 & y \end{matrix} ⎤ ⎥ ⎦$ and $A \cdot A^{T} = I$ , then find the value of $x + y$ .
$[3]$

(c) If $A = [\begin{matrix} 2 & 5 1 & 3 \end{matrix}], B = [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$ and $I$ is Identity matrix of same order and $A^{t}$ is the transpose of matrix $A$ find $A^{t} . B + B I$ .
$[4]$

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & - 1 & 1 2 & 1 & - 3 1 & 1 & 1 \end{matrix} ⎤ ⎥ ⎦

and

10 B = ⎡ ⎢ ⎣ \begin{matrix} 4 & 2 & 2 - 5 & 0 & α 1 & - 2 & 3 \end{matrix} ⎤ ⎥ ⎦

. If

B

is the inverse

o f A

, then

α

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If A=[2513],B=[4−2−13] and I is Identity matrix of same order and At is the transpose of matrix A find At.B+BI [4 Marks]

A=[2513] ∴At=[2153] [1 Mark] At.B=[2153][4−2−13] =[2×4+1×(−1)2×(−2)+1×33×4+3×(−1)5×(−2)+3×3] =[7−117−1] [1 Mark] B.I=[4−2−13][1001] =[4−2−13] [1 Mark] ∴At.B+BI=[7−117−1]+[4−2−13] =[11−3162] [1 Mark]

If $A = [\begin{matrix} 2 & 5 1 & 3 \end{matrix}], B = [\begin{matrix} 4 & - 2 - 1 & 3 \end{matrix}]$ and $I$ is Identity matrix of same order and $A^{t}$ is the transpose of matrix $A$ find $A^{t} . B + B I$ [4 Marks]