Question:
if A = $\begin{bmatrix}1&3\\ 3&4\end{bmatrix}$ and $A^2 - kA - 5I = 0$, then $k$ =
if A = $\begin{bmatrix}1&3\\ 3&4\end{bmatrix}$ and $A^2 - kA - 5I = 0$, then $k$ =
Updated On: Jul 6, 2022
- $5$
- $3$
- $7$
- None of these
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The Correct Option is A
Solution and Explanation
Given $A^{2} -kA- 5I= 0$
$\Rightarrow kA=A^{ 2 }-5I $
$\Rightarrow kA=\begin{bmatrix}1&3\\ 3&4\end{bmatrix}\begin{bmatrix}1&3\\ 3&4\end{bmatrix}-5\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$
=$\begin{bmatrix}10&15\\ 15&25\end{bmatrix}-\begin{bmatrix}5&0\\ 0&5\end{bmatrix}=\begin{bmatrix}5&15\\ 15&20\end{bmatrix}=5\begin{bmatrix}1&3\\ 3&4\end{bmatrix}=5A$
$\Rightarrow kA = 5A \therefore k=5$
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.