JEE Mathematics 2023 Question with Solution

Given: The matrix expression involves $M$ and asks for the inverse of $A M^{2023} A^T$.

Find: The correct option for $\left(A M^{2023} A^T\right)^{-1}$.

From the solution,

$$M = \begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix}$$

The powers follow the pattern

$$M^2 = \begin{pmatrix} 1 & 2i \\ 0 & 1 \end{pmatrix}, \quad M^3 = \begin{pmatrix} 1 & 3i \\ 0 & 1 \end{pmatrix}, \quad M^n = \begin{pmatrix} 1 & ni \\ 0 & 1 \end{pmatrix}$$

Hence,

$$M^{2023} = \begin{pmatrix} 1 & 2023i \\ 0 & 1 \end{pmatrix}$$

the solution further states that

$$A M^{2023} A^\dagger = \begin{pmatrix} 1 & 2023i \\ 0 & 1 \end{pmatrix}$$

Therefore its inverse is

$$\left(A M^{2023} A^\dagger\right)^{-1} = \begin{pmatrix} 1 & -2023i \\ 0 & 1 \end{pmatrix}$$

This matrix matches Option C among the given images.

Note: The answer key says (4) and the final line of the working says Option (3), while the heading says The Correct Option is C. The matrix derived in the working matches Option C, so the answer is C.

Given:

$$M = \begin{pmatrix} 1 & i \\ 0 & 1 \end{pmatrix}$$

Find: Use the power pattern and identify the inverse matrix.

Since $M$ is an upper triangular matrix of the form

$$\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$$

its powers add the upper-right entry repeatedly. So,

$$M^n = \begin{pmatrix} 1 & ni \\ 0 & 1 \end{pmatrix}$$

Thus,

$$M^{2023} = \begin{pmatrix} 1 & 2023i \\ 0 & 1 \end{pmatrix}$$

The inverse of a matrix of the form

$$\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$$

is

$$\begin{pmatrix} 1 & -x \\ 0 & 1 \end{pmatrix}$$

Therefore,

$$\left(A M^{2023} A^T\right)^{-1}$$

corresponds to

$$\begin{pmatrix} 1 & -2023i \\ 0 & 1 \end{pmatrix}$$

which is Option C according to the solution working.