(D) Given that $(A-3 I)(A-5 I)=O$. Expanding the expression,we get: $A^2 - 5A - 3A + 15I = O$ $A^2 - 8A + 15I = O$ $A^2 + 15I = 8A$ Multiplying both s

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(D) Given that $(A-3 I)(A-5 I)=O$. Expanding the expression,we get: $A^2 - 5A - 3A + 15I = O$ $A^2 - 8A + 15I = O$ $A^2 + 15I = 8A$ Multiplying both sides by $A^{-1}$,we get: $A + 15A^{-1} = 8I$ Dividing the entire equation by $2$: $\frac{1}{2}A + \frac{15}{2}A^{-1} = 4I$ Comparing this with the given equation $\alpha A + \beta A^{-1} = 4I$,we identify: $\alpha = \frac{1}{2}$ and $\beta = \frac{15}{2}$ Therefore,$\alpha + \beta = \frac{1}{2} + \frac{15}{2} = \frac{16}{2} = 8$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Adjoint and inverse of matrices

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Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$,where $I=I_3$ and $O=O_3$. Here $O_3$ represents the zero matrix of order $3$ and $I_3$ is the identity matrix of order $3$. If $\alpha A+\beta A^{-1}=4 I$,then $\alpha+\beta$ is equal to:

MHT CET 2024 All MHT CET

A
$13$
B
$7$
C
$12$
D
$8$

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3 and 4 .Determinants and Matrices

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Adjoint and inverse of matrices

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$A$ and $B$ are $n$-rowed square matrices such that $AB = O$ and $B$ is non-singular. Then:

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If $A=\begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$,$B=\begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix}$,$i=\sqrt{-1}$,and $Q=A^{T}BA$,then the inverse of the matrix $AQ^{2021}A^{T}$ is equal to:

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If for the matrix $A$,${A^3} = I$,then ${A^{-1}} = $

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If $A$ and $B$ are non-singular matrices,then which of the following is true?

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If the inverse matrix of $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ is $A^{-1} = \begin{bmatrix} a & 3/11 \\ 1/11 & b \end{bmatrix}$, then $a+b=$ . . . . . . .

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Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$,where $I=I_3$ and $O=O_3$. Here $O_3$ represents the zero matrix of order $3$ and $I_3$ is the identity matrix of order $3$. If $\alpha A+\beta A^{-1}=4 I$,then $\alpha+\beta$ is equal to:

Similar Questions

$A$ and $B$ are $n$-rowed square matrices such that $AB = O$ and $B$ is non-singular. Then:

If $A=\begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix}$,$B=\begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix}$,$i=\sqrt{-1}$,and $Q=A^{T}BA$,then the inverse of the matrix $AQ^{2021}A^{T}$ is equal to:

If for the matrix $A$,${A^3} = I$,then ${A^{-1}} = $

If $A$ and $B$ are non-singular matrices,then which of the following is true?

If the inverse matrix of $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ is $A^{-1} = \begin{bmatrix} a & 3/11 \\ 1/11 & b \end{bmatrix}$, then $a+b=$ . . . . . . .

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