Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & -i \\ 0 & 1\end{array}\right]$, where $i=\sqrt{-1}$ If $M = A ^{ T } B A$, then the inverse of the matrix $AM ^{2023} A ^{ T }$ is
- $\begin{bmatrix}1 & -2023 i \\ 0 & 1\end{bmatrix}$
- $\begin{bmatrix}1 & 0 \\ 2023 i & 1\end{bmatrix}$
- $\begin{bmatrix}1 & 2023 i \\ 0 & 1\end{bmatrix}$
- $\begin{bmatrix}1 & 0 \\ -2023 i & 1\end{bmatrix}$
The Correct Option is C
Solution and Explanation
We compute \( M \) as \( M = A^\dagger B A \). The adjoint (conjugate transpose) of \( A \), \( A^\dagger \), and the product \( A^\dagger B A \) lead to a specific form of \( M \):
\[ M = \begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix}. \]
To find \( M^{2023} \), we observe the pattern of powers of \( M \):
\[ M^2 = \begin{pmatrix} 1 & 2i \\ 0 & 1 \end{pmatrix}, \quad M^3 = \begin{pmatrix} 1 & 3i \\ 0 & 1 \end{pmatrix}, \quad M^n = \begin{pmatrix} 1 & ni \\ 0 & 1 \end{pmatrix}. \]
Thus:
\[ M^{2023} = \begin{pmatrix} 1 & 2023i \\ 0 & 1 \end{pmatrix}. \]
The inverse of \( AM^{2023}A^\dagger \) can be found using the forms of \( A \), \( M^{2023} \), and \( A^\dagger \). The computation shows that:
\[ AM^{2023}A^\dagger = \begin{pmatrix} 1 & 2023i \\ 0 & 1 \end{pmatrix}. \]
Therefore, its inverse is:
\[ \text{Inverse} = \begin{pmatrix} 1 & -2023i \\ 0 & 1 \end{pmatrix}. \]
Thus, the correct answer is Option (3).
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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.
Types of Matrices
There are many types of matrices and are further categorized on the basis of the value of their elements, their order, number of rows and columns, etc.
- Row Matrix - A = [aij]1×n
- Column Matrix - A = [aij]mxn
- Zero or Null Matrix - A = [aij]mxn, {aij = 0}
- Singleton Matrix - A = [aij]mxn, {m = n =1}
- Horizontal Matrix - [aij]mxn, {n > m}
- Vertical Matrix - [aij]mxn, {m > n}
- Square Matrix - [aij]mxn, {m = n}
- Diagonal Matrix - A = [aij], {i ≠ j}
- Scalar Matrix - A = [aij]mxn, {aij = {0, I = j} {k, I ≠ j}}, {k = constant}.
- Identity (Unit) Matrix - A = [aij]mxn, {aij = {1, i=j, {1, i≠j}
- Equal Matrix - A = [aij]mxn and B = [bij]rxs, {aij = bij, m = r, and n = s}
- Triangular Matrices - Either upper triangular (aij = 0, when i > j) or lower triangular (aij = 0 when i < j)
- Singular Matrix - |A| = 0
- Non-Singular Matrix - |A| ≠ 0
- Symmetric Matrices - A = [aij], {aij = aji}
- Skew-Symmetric Matrices - A = [aij], {aij = aji}
- Hermitian Matrix - A = Aθ
- Skew – Hermitian Matrix - Aθ = -A
- Orthogonal Matrix - A AT = In = AT A
- Idempotent Matrix - A2 = A
- Involuntary Matrix - A2 = I, A-1 = A
- Nilpotent Matrix - ∃ p ∈ N such that AP = 0
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