Question:
If $A = \begin{bmatrix}i&0\\ 0&-i\end{bmatrix}, B = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$ and $ C= \begin{bmatrix}0&i\\ i&0\end{bmatrix} $ then $A^{2} = B^{2} =C^{2} $ is equal to :
If $A = \begin{bmatrix}i&0\\ 0&-i\end{bmatrix}, B = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$ and $ C= \begin{bmatrix}0&i\\ i&0\end{bmatrix} $ then $A^{2} = B^{2} =C^{2} $ is equal to :
Updated On: Jul 6, 2022
- $I^2$
- I
- - I
- 2I
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Let $A = \begin{bmatrix}i&0\\ 0&-i\end{bmatrix}, B = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$ and $ C= \begin{bmatrix}0&i\\ i&0\end{bmatrix} $
$A. A= \begin{pmatrix}i&0\\ 0&-i\end{pmatrix} \begin{pmatrix}i&0\\ 0&-1\end{pmatrix} $
$= \begin{pmatrix}i^{2}&0\\ 0&i^{2}\end{pmatrix} = \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} = - I$
and $ B.B = \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix} $
$ = \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} = -I$
$ C.C = \begin{pmatrix}0&i\\ i&0\end{pmatrix}\begin{pmatrix}0&i\\ i&0\end{pmatrix} = \begin{pmatrix}i^{2}&0\\ 0&i^{2}\end{pmatrix} = -I $
Hence, $A^{2} = B^{2} = C^{2} = - I $
Was this answer helpful?
0
0
Top Questions on Matrices
- Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:
- Given \[ A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix} \] find \( AB \). Hence, solve the system of linear equations: \[ x - y + z = 4, \] \[ x - 2y - 2z = 9, \] \[ 2x + y + 3z = 1. \]
- If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix} \] then find \( A^{-1} \). Hence, solve the system of linear equations: \[ x - 2y = 10, \] \[ 2x - y - z = 8, \] \[ -2y + z = 7. \]
- Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix. - If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?
View More Questions
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.