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Multiplication of Matrices

Let A be a 2×...

Question

Let $A$ be a $2 \times 2$ matrix with real entries, Let I be the $2 \times 2$ identity matrix. Denote by $t r (A),$ the sum of diagonal entries of $A$ . Assume that $A^{2} = I$
Statement $1 :$ If $A \neq I$ and $A \neq - I$ , then det $A = - 1$
Statement $2 :$ If $A \neq I$ and $A \neq - I$ , then $t r (A) \neq 0,$ then which of the following is correct

Both statement

1

2

are true but statement

2

is not a correct explanation for statement

1

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Statment

1

is false, statement

2

is true.

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Both statement

1

2

are true and statement

2

is a correct explanation for statement

1

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Statement

1

is true, statement

2

is false

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Solution

The correct option is D Statement $1$ is true, statement $2$ is false
Let $A = [\begin{matrix} a & b c & d \end{matrix}]$ .
Then $A^{2} = [\begin{matrix} a^{2} + b c & a b + b d a c + d c & b c + d^{2} \end{matrix}] = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$
Hence
$det (A) = ∣ ∣ ∣ \begin{matrix} \sqrt{1 - b c} & b c & - \sqrt{1 - b c} \end{matrix} ∣ ∣ ∣ = - 1 + b c - b c = - 1$
Here, we can see $T r (A) = 0$ so, 2nd statement is false only.

Suggest Corrections

Similar questions

Q. Let

A

be a

2 \times 2

matrix with real entries, Let I be the

2 \times 2

identity matrix. Denote by

t r (A),

the sum of diagonal entries of

A

. Assume that

A^{2} = I

Statement

1 :

A \neq I

and

A \neq - I

, then det

A = - 1

Statement

2 :

A \neq I

and

A \neq - I

, then

t r (A) \neq 0,

then which of the following is correct

Q. Assertion :Let

A

be a

2 \times 2

matrix with real entries. Let

I

be the

2 \times 2

identity matrix. Denote by

t r (A)

, the sum of diagonal entries of

A

. Assume that

A^{2} = I

A \neq I

and

A \neq - I

, then

d e t (A) = - 1

Reason: If

A \neq I

and

A \neq - I

, then

t r (A) \neq 0

Q. Let A be a

2 \times 2

matrix with real entries. Let I be the

2 \times 2

identity matrix. Denote by tr(A), the sum of diagonal entries of

A

. Assume that

A^{2} =

I.
Statement-l: If

A \neq I

and

A \neq - I

, then

det A = - 1

.
Statement-2: If

A \neq I

and

A \neq - I

, then

t_{1} {A) \neq 0

Q. Let A be a

2 \times 2

matrix with non-zero entries and let

A^{2} = I

, where I is

2 \times 2

identity matrix. Define Tr(A)

=

sum of diagonal elements of A and

| A | =

determinant of matrix A.
Statement-1 Tr(A)

= 0

Statement-2:

| A | = 1

Q. Let

A

be a

2 \times 2

real matrix with entries from

{0, 1}

and

| A | \neq 0

. Consider the following two statements:

(P) If

A \neq I_{2},

then

| A | = - 1

(Q) If

| A | = 1

, then

tr (A) = 2

,
where

I_{2}

denotes

2 \times 2

identity matrix and

tr (A)

denotes the sum of the diagonal entries of

A

. Then :

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Multiplication of Matrices

Standard XIII Mathematics

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Let A be a 2×2 matrix with real entries, Let I be the 2×2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2=I Statement 1: If A≠I and A≠−I, then detA=−1 Statement 2: If A≠I and A≠−I, then tr(A)≠0, then which of the following is correct

The correct option is D Statement 1 is true, statement 2 is falseLet A=[abcd]. Then A2=[a2+bcab+bdac+dcbc+d2]=[1001] Hence det(A)=∣∣∣√1−bcbc−√1−bc∣∣∣=−1+bc−bc=−1 Here, we can see Tr(A)=0 so, 2nd statement is false only.