For $A=\begin{bmatrix}1&2\\2&1\end{bmatrix}$, verify that $A$ is symmetric, and find $A^{-1}$.

$A=\begin{bmatrix}1&2\\2&1\end{bmatrix}$, so $A^T=\begin{bmatrix}1&2\\2&1\end{bmatrix}=A$; since $A^T=A$, $A$ is symmetric. For the inverse, $\det A=1(1)-2(2)=1-4=-3\neq0$, so $A$ is invertible, and $A^{-1}=\dfrac{1}{-3}\begin{bmatrix}1&-2\\-2&1\end{bmatrix}=\begin{bmatrix}-1/3&2/3\\2/3&-1/3\end{bmatrix}$. Checking: $AA^{-1}$ position $(1,1)$: $1(-1/3)+2(2/3)=(-1+4)/3=1$; position $(1,2)$: $1(2/3)+2(-1/3)=(2-2)/ …

For A= 12; 21 , verify that A is symmetric, and find A⁻¹.

Miscellaneous · Q32

A=\begin{bmatrix}1&2\\2&1\end{bmatrix}

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$A=\begin{bmatrix}1&2\2&1\end{bmatrix}$ , so $A^T=\begin{bmatrix}1&2\2&1\end{bmatrix}=A$ ; since $A^T=A$ , $A$ is symmetric. For the inverse, $\det A=1(1)-2(2)=1-4=-3 eq0$ , so $A$ is invertible, and $A^{-1}=\dfrac{1}{-3}\begin{bmatrix}1&-2\-2&1\end{bmatrix}=\begin{bmatrix}-1/3&2/3\2/3&-1/3\end{bmatrix}$ . Checking: $AA^{-1}$ position $(1,1)$ : $1(-1/3)+2(2/3)=(-1+4)/3=1$ ; position $(1,2)$ : $1(2/3)+2(-1/3)=(2-2)/ …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Matrices A and B will be inverse of each other only if$
(a) AB = BA
(b) AB = BA = O
(c) AB = O, BA = I
(d) AB = BA = I
›Reveal solutionSolution

Two square matrices are inverses of each other exactly when both products give the identity matrix.

By definition, if $A$ and $B$ are square matrices of the same order such that $AB=BA=I$ (the identity matrix), then $B$ is called the inverse of $A$ (written $B=A^{-1}$ ) and $A$ is the inverse of $B$ . A si …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. If the matrix (4 -k; -2 3) has no inverse matrix, then the value of k is$
(a) 6
(b) -6
(c) 12
(d) -12
›Reveal solutionSolution

A matrix has no inverse exactly when its determinant is zero.

A square matrix is invertible only if it is non-singular, i.e. $\det\ne 0$ . "No inverse matrix" means the matrix IS singular, so we need $\det = 0$ .

…

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CBSE 2024Set ANNUAL1 markMCQ
$A$
(a) $AB=BA$
(b) $AB=BA=0$
(c) $AB=0, BA=I$
(d) $AB=BA=I$
›Reveal solutionSolution

By definition, A and B are inverses of each other exactly when their product (in either order) gives the identity matrix.

A square matrix $B$ is the inverse of $A$ if $AB=BA=I$ , where $I$ is the identity matrix of the same order. This is the standard definition of ma …

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CBSE 2023Set ANNUAL1 markMCQ
$Q. Matrices A and B will be inverse of each other only if-$
(a) $AB = BA$
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(c) $AB = 0, BA = I$
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›Reveal solutionSolution

Inverse of a matrix is defined by the identity relation on both sides.

By definition, if $A$ is a square matrix and there exists a matrix $B$ of the same order such that $AB=BA=I$ (the identity matrix), then $B$ is called the inverse of $A$ (written $B=A^{-1}$ ), and $A$ is the i …

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Q.For $A=\begin{bmatrix}1&2\\2&1\end{bmatrix}$ , verify that $A$ is symmetric, and find $A^{-1}$ .

Concept understanding — Invertible Matrices and Uniqueness of Inverse

Q.For A=[1221]A=\begin{bmatrix}1&2\\2&1\end{bmatrix}A=[12​21​], verify that AAA is symmetric, and find A−1A^{-1}A−1.

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Q.For $A=\begin{bmatrix}1&2\\2&1\end{bmatrix}$ , verify that $A$ is symmetric, and find $A^{-1}$ .

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