Question:
If $ A=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}$, I is the unit matrix of order 2 and a, b are arbitrary constants, then $(aI + bA)^2$ is equal to
If $ A=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}$, I is the unit matrix of order 2 and a, b are arbitrary constants, then $(aI + bA)^2$ is equal to
Updated On: Jul 6, 2022
- $a^2I + abA$
- $a^2I + 2abA$
- $a^2I + b^2A$
- None of these
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The Correct Option is B
Solution and Explanation
$\left(aI+bA\right)^{2}=a^{2}I^{2}+b^{2}A^{2}+2ab\,AI$
$=a^{2}I^{2}+b^{2}\,A^{2}+2abA$
But $A^{2}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix} \therefore \left(aI+bA\right)^{2}=a^{2}I+2abA.$
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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.