Question:
If $ A= \begin{bmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ x & x & 1 \\ \end{bmatrix} $ and $ I= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} , $ then $ {{A}^{3}}-4{{A}^{2}}+3A+I $ is equal to
If $ A= \begin{bmatrix} 1 & 0 & 0 \\ x & 1 & 0 \\ x & x & 1 \\ \end{bmatrix} $ and $ I= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} , $ then $ {{A}^{3}}-4{{A}^{2}}+3A+I $ is equal to
Updated On: Jun 7, 2024
- $ 3I $
- $ I $
- $ -I $
- $ -2I $
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The Correct Option is B
Solution and Explanation
Given, $ A=\left| \begin{matrix} 1 & 0 & 0 \\ x & 1 & 0 \\ x & x & 1 \\ \end{matrix} \right|\Rightarrow A=1 $
$ \therefore $ $ {{A}^{3}}-4{{A}^{2}}+3A+I={{(1)}^{3}}-4{{(1)}^{3}}+3(1)+I $
$=I $
$ \therefore $ $ {{A}^{3}}-4{{A}^{2}}+3A+I={{(1)}^{3}}-4{{(1)}^{3}}+3(1)+I $
$=I $
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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.