(D) Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.Given $A^2 = I$,we have:$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a & b

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Statement $-1$ is true,Statement $-2$ is false.

(D) Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.Given $A^2 = I$,we have:$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$\begin{bmatrix} a^2 + bc & b(a+d) \\ c(a+d) & bc + d^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$From $b(a+d) = 0$,since $b \neq 0$,we get $a+d = 0$,which implies $d = -a$.Thus,$tr(A) = a + d = a - a = 0$. So,Statement $-1$ is true.Now,$|A| = ad - bc = a(-a) - bc = -(a^2 + bc)$.From the matrix multiplication,$a^2 + bc = 1$,so $|A| = -1$.Therefore,Statement $-2$ is false.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Mix Examples-Determinants and Matrices

Medium

Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2 = I$,where $I$ is the $2 \times 2$ identity matrix. Define $tr(A) = \text{sum of diagonal elements of } A$ and $|A| = \text{determinant of matrix } A$.
Statement $-1: tr(A) = 0$
Statement $-2: \det(A) = 1$

AIEEE 2010 All AIEEE

A
Statement $-1$ is false,Statement $-2$ is true.
B
Statement $-1$ is true,Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$.
C
Statement $-1$ is true,Statement $-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$.
D
Statement $-1$ is true,Statement $-2$ is false.

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Let $M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}$,where $\alpha$ is a non-zero real number and $N = \sum_{k=1}^{49} M^{2k}$. If $(I - M^2)N = -2I$,then the positive integral value of $\alpha$ is

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For the matrices $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}$, if $(A^{15}+B)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, then among the following which one is true?

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Let $P$ and $Q$ be $3 \times 3$ matrices such that $P \neq Q$. If $P^3 = Q^3$ and $P^2Q = Q^2P$,then the determinant $\det(P^2 + Q^2)$ is equal to:

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Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2 = I$,where $I$ is the $2 \times 2$ identity matrix. Define $tr(A) = \text{sum of diagonal elements of } A$ and $|A| = \text{determinant of matrix } A$.Statement $-1: tr(A) = 0$Statement $-2: \det(A) = 1$

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The maximum value of $f(x) = \left|\begin{array}{ccc} \sin^{2} x & 1+\cos^{2} x & \cos 2x \\ 1+\sin^{2} x & \cos^{2} x & \cos 2x \\ \sin^{2} x & \cos^{2} x & \sin 2x \end{array}\right|, x \in R$ is:

The equation $\left| \begin{array}{ccc} (1+x)^2 & (1-x)^2 & -(2+x^2) \\ 2x+1 & 3x & 1-5x \\ x+1 & 2x & 2-3x \end{array} \right| + \left| \begin{array}{ccc} (1+x)^2 & 2x+1 & x+1 \\ (1-x)^2 & 3x & 2x \\ 1-2x & 3x-2 & 2x-3 \end{array} \right| = 0$

Let $M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}$,where $\alpha$ is a non-zero real number and $N = \sum_{k=1}^{49} M^{2k}$. If $(I - M^2)N = -2I$,then the positive integral value of $\alpha$ is

For the matrices $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix}$, if $(A^{15}+B)\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, then among the following which one is true?

Let $P$ and $Q$ be $3 \times 3$ matrices such that $P \neq Q$. If $P^3 = Q^3$ and $P^2Q = Q^2P$,then the determinant $\det(P^2 + Q^2)$ is equal to:

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Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2 = I$,where $I$ is the $2 \times 2$ identity matrix. Define $tr(A) = \text{sum of diagonal elements of } A$ and $|A| = \text{determinant of matrix } A$.
Statement $-1: tr(A) = 0$
Statement $-2: \det(A) = 1$