Question:
Let I denote the $3 \times 3$ identity matrix and P be a matrix obtained by rearranging the columns of I. Then
Let I denote the $3 \times 3$ identity matrix and P be a matrix obtained by rearranging the columns of I. Then
Updated On: Jul 6, 2022
- There are six distinct choices for P and det (P) = 1
- There are six distinct choices for P and det (P) = $\pm$ 1
- There are more than one choices for P and some of them are invertible
- There are more than one choices for P and $P^1$ = I is each choice
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The Correct Option is B
Solution and Explanation
$I_{3 \times3} = \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
3 different columns can be arranged in 3! i.e., 6 ways. In each case, if there are even number of interchanges of columns, determinant remains 1 and for odd number of interchange determinant takes the negative value i.e., -1.
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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.