Matrix \(\left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{ - 3}\\ 2&1&{ - 2}\\ 3&2&1 \end{array}} \right]\) is ___

Concept:  

A symmetric matrix 

• It is a square matrix A of size n × n when the square matrix is equal to the transposed form of that matrix, that is, A^T = A.
• If A = [a_ij]_n×n is the symmetric matrix, then a_ij = a_ji for 1 ≤ i ≤ n, and 1 ≤ j ≤ n.

A skew-symmetric matrix 

• It is a square matrix A of size n × n when the square matrix is equal to the transposed form of that matrix, that is, A^T = -A.
• Diagonal elements of the skew-symmetric is zero.

If A = [a_ij]_n×n is the skew-symmetric matrix, then a_ij = -a_ji for 1 ≤ i ≤ n, and 1 ≤ j ≤ n.

A square matrix is called a singular matrix when its determinant is equal to 0.

A square matrix is called a non-singular matrix when its determinant is not equal to 0.

Calculation:

Given:

The given matrix is A = \(\begin{vmatrix} 1& -2& -3 \\ 2& 1& -2 \\ 3& 2 & 1 \\ \end{vmatrix}\).

Transpose of the matrix is A^t = \(\begin{vmatrix} 1& 2& 3 \\ -2& 1& 2 \\ -3& -2 & 1 \\ \end{vmatrix}\).

Since A^t ≠ A, hence the matrix A is not symmetric.

Since diagonal elements of the matrix are not zero, then matrix A is not skew-symmetric.

The determinant ∆ of the matrix is given by,

∆ = 1(1(1) - 2(-2)) - (-2)(1(2) - 3(-2)) + (-3)(2(2) - 1(3))

\(\Rightarrow\) ∆ = 1(1 + 4) + 2(2 + 6) - 3(4 - 3)

\(\Rightarrow\) ∆ = 5 + 16 - 3

\(\Rightarrow\) ∆ = 18

The determinant of the matrix is not equal to zero, hence it is non-singular.

Hence, the correct answer is option 4.