Question:
If $ A= \begin{bmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{bmatrix} , $ then $ {{A}^{4}} $ is equal to
If $ A= \begin{bmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{bmatrix} , $ then $ {{A}^{4}} $ is equal to
Updated On: Jun 6, 2022
- $27 \,A$
- $81\, A$
- $243\, A$
- $729\, A$
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The Correct Option is D
Solution and Explanation
Given, $ A=\left[ \begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{matrix} \right] $
$ \therefore $ $ A=3\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right| $
$ \therefore $ $ {{A}^{2}}=3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right].3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right] $
$=9\left[ \begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{matrix} \right]=9A $
$ \therefore $ $ {{A}^{4}}={{A}^{2}}.{{A}^{2}} $
$=9A.9A=81{{A}^{2}}=81.9A $
$=729A $
$ \therefore $ $ A=3\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right| $
$ \therefore $ $ {{A}^{2}}=3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right].3\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{matrix} \right] $
$=9\left[ \begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{matrix} \right]=9A $
$ \therefore $ $ {{A}^{4}}={{A}^{2}}.{{A}^{2}} $
$=9A.9A=81{{A}^{2}}=81.9A $
$=729A $
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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.