Adjoint and Inverse of a Matrix - Gujarat Board (GSEB) STD 12 Science Maths Questions

Q 1M.C.Q (1 Marks)1 Mark

Let $\text{A}=\begin{bmatrix} 1 & 2 \\ 3 & -5 \end{bmatrix}\text{ and B}=\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ and X be a matrix such that A = BX, then X is equal to:

$\frac{1}{2}\begin{bmatrix} 2 & 4 \\ 3 & -5 \end{bmatrix}$
$\frac{1}{2}\begin{bmatrix} -2 & 4 \\ 3 & 5 \end{bmatrix}$
$\begin{bmatrix} 2 & 4 \\ 3 & -5 \end{bmatrix}$
None of these.

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Q 2M.C.Q (1 Marks)1 Mark

Let $\text{A}=\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}$ be such that A^-1 = kA, then k equals:

$19$
$\frac{1}{19}$
$-19$
$-\frac{1}{19}$

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Q 3M.C.Q (1 Marks)1 Mark

If x, y, z are non-zero real numbers, then the inverse, then the inverse of the matrix $\begin{bmatrix}\text{x} & 0 & 0\\ 0 & \text{y} & 0 \\ 0 & 0 & \text{z}\end{bmatrix}$, is:

$\begin{bmatrix}\text{x}^{-1} & 0 & 0\\ 0 & \text{y}^{-1} & 0 \\ 0 & 0 & \text{z}^{-1}\end{bmatrix}$
$\text{xyz}\begin{bmatrix}\text{x}^{-1} & 0 & 0\\ 0 & \text{y}^{-1} & 0 \\ 0 & 0 & \text{z}^{-1}\end{bmatrix}$
$\frac{1}{\text{xyz}}\begin{bmatrix}\text{x} & 0 & 0\\ 0 & \text{y} & 0 \\ 0 & 0 & \text{z}\end{bmatrix}$
$\frac{1}{\text{xyz}}\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

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Q 4M.C.Q (1 Marks)1 Mark

If $\text{A}=\begin{bmatrix} 3 & 4 \\ 2 & 4 \end{bmatrix},\text{B}=\begin{bmatrix} -2 & -2 \\ 0 & -1 \end{bmatrix}$ then (A + B)^-1 =

Is A akew-symmetric matrix.
A^-1 + B^-1
Does not exist.
None of these.

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Q 5M.C.Q (1 Marks)1 Mark

If A is an invertible matrix of order 3, then which of the following is not true:

$|\text{adj A}|=|\text{A}|^2$
$(\text{A}^{-1})^{-1}=\text{A}$
If BA = CA, than $\text{B}\neq\text{C},$ where B and C are square matrices of order 3
$(\text{AB})^{-1}=\text{B}^{-1}\text{A}^{-1},$ where $\text{B}\neq\big[\text{b}_{\text{ij}}\big]_{3\times3}\text{ and |B|}\neq0$

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Q 62 Marks2 Marks

Find the adjoint of the following matrices:

$\begin{bmatrix}1 & \frac{\tan\alpha}{2} \\ -\frac{\tan\alpha}{2} & 1 \end{bmatrix}$

Verify that (adjoint A) A = |A|I = A (adjoint A) for the above matrices.

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Q 72 Marks2 Marks

If A is an invertible matrix such that |A^-1| = 2, find the value of |A|.

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Q 82 Marks2 Marks

If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.

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Q 92 Marks2 Marks

If A is a square matrix of order 3 such that |A| = 2, then write the value of adj (adj A).

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Q 102 Marks2 Marks

If A is a square matrix of order 3 such that |A| = 5, write the value of |adj A|.

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Q 113 Marks3 Marks

Find the inverse of the following matrices:

$\begin{bmatrix}2 & 5 \\ -3 & 1 \end{bmatrix}$

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Q 123 Marks3 Marks

Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$ and

$\text{G}(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$

Show that

$\big[\text{G}(\beta)\big]^{-1}=\text{G}(-\beta)$

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Q 133 Marks3 Marks

If $\text{A}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\text{ and A (adj A =)}\begin{bmatrix} \text{k} & 0 \\ 0 & \text{k} \end{bmatrix},$ then find the value of k.

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Q 143 Marks3 Marks

If $\text{A}=\begin{bmatrix}2 & 3 \\ 5 & -2 \end{bmatrix}$ be sech that A^-1 = kA, then find the value of k.

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Q 153 Marks3 Marks

If $\text{adj A}=\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \text{and B}=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix},$ find adj AB.

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Q 164 Marks4 Marks

If $\text{A}=\frac{1}{9}\begin{bmatrix}-8 & 1 & 4\\4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix},$ prove that A^-1 = A³.

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Q 174 Marks4 Marks

Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$ and

$\text{G}(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$

Show that

$\big[\text{F}(\alpha)\big]^{-1}=\text{F}(-\alpha)$

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Q 184 Marks4 Marks

If $\text{A}=\begin{bmatrix}4 & 3 \\ 2 & 5 \end{bmatrix},$ find x and y such that A² = zA + yI = 0. Hence, evaluate A^-1.

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Q 194 Marks4 Marks

Find the inverse of the following matrices:

$\begin{bmatrix}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$

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Q 204 Marks4 Marks

Find the inverse of the following matrices by using elementry row transformation:

$\begin{bmatrix}3 & 10 \\ 2 & 7 \end{bmatrix}$

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