If $A = [a_{ij}]_{2×2}$ where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ and $I$ is the identity matrix of order 2, then $(A^2 - 3A + 4I)$ is

(A) Symmetric Matrix

(B) Skew-symmetric Matrix

(C) Non-singular Matrix

(D) Square Matrix

Choose the correct answer from the options given below:

The condition $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$ tells us:

When row number ≠ column number → element is 1

When row number = column number → element is 0

For the 2×2 matrix:

Position (1,1): row 1 = column 1 → 0

Position (1,2): row 1 ≠ column 2 → 1

Position (2,1): row 2 ≠ column 1 → 1

Position (2,2): row 2 = column 2 → 0

$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

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$A^2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Row 1, Column 1: $(0)(0) + (1)(1) = 1$

Row 1, Column 2: $(0)(1) + (1)(0) = 0$

Row 2, Column 1: $(1)(0) + (0)(1) = 0$

Row 2, Column 2: $(1)(1) + (0)(0) = 1$

$A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$A^2 = I$

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$A^2 - 3A + 4I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - 3\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} + 4\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 0 & 3 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}$

$= \begin{bmatrix} 1-0+4 & 0-3+0 \\ 0-3+0 & 1-0+4 \end{bmatrix}$

$= \begin{bmatrix} 5 & -3 \\ -3 & 5 \end{bmatrix}$

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Checking option (A) - Symmetric Matrix:

A matrix is symmetric when $a_{ij} = a_{ji}$

Element at (1,2) is $-3$

Element at (2,1) is $-3$

Since $-3 = -3$, the matrix is symmetric.

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Checking option (B) - Skew-symmetric Matrix:

A matrix is skew-symmetric when $a_{ij} = -a_{ji}$ for all $i,j$

Element at (1,2) is $-3$

Element at (2,1) is $-3$

Check: $-3 \neq -(-3) = 3$

The matrix is not skew-symmetric.

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Checking option (C) - Non-singular Matrix:

A matrix is non-singular when its determinant ≠ 0

$\text{Determinant} = (5)(5) - (-3)(-3)$

$= 25 - 9$

$= 16 \neq 0$

The matrix is non-singular.

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Checking option (D) - Square Matrix:

A matrix is square when it has equal number of rows and columns.

The matrix is 2×2 (2 rows, 2 columns).

The matrix is a square matrix.

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The matrix $(A^2 - 3A + 4I) = \begin{bmatrix} 5 & -3 \\ -3 & 5 \end{bmatrix}$ is:

(A) Symmetric Matrix ✓

(C) Non-singular Matrix ✓

(D) Square Matrix ✓

Properties (A), (C), and (D) are all correct.