Singular Matrix - Practice Questions & MCQ

The determinant of a matrix A is a number which is calculated from the matrix. For determinant to exist, matrix A must be a square matrix. Determinant of matrix is denoted by det A or |A|. 

For $2 \times 2$ matrices

$
\mathrm{A}=\left[\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right]
$

then $\operatorname{det} \mathrm{A}$ is :

$
|\mathrm{A}|=\left|\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right|=\mathrm{a}_1 \times \mathrm{b}_2-\mathrm{a}_2 \times \mathrm{b}_1
$
For a $3 \times 3$ matrix determinant can be calculated in the following way :
let $\mathrm{A}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$
then we find $\operatorname{det} \mathrm{A}$ in following way

$
|A|=a_1\left(b_2 \cdot c_3-b_3 \cdot c_2\right)-a_2\left(b_1 \cdot c_3-c_1 b_3\right)+a_3\left(b_1 c_2-b_2 c_1\right)
$

This same process we follow to evaluate the determinant of the matrix of any order. Notice that we start first term with +ve sign and then 2nd with -ve sign and 3rd again +ve sign, this sign sequence is followed for any order of matrix.

This whole process is row dependent, same process can be done using column, means we can select element along column and delete their row and column and compute the determinant of left out matrix and then multiply it with the element which we select. And we will get the same result as we get while doing the whole process along row.

Singular and non-singular matrix:

A square matrix is called singular matrix if its determinant is 0 otherwise it is called non-singular matrix. Let say A is a square matrix then it is singular if |A| = 0, otherwise, it will be non-singular if |A| ≠ 0.

Properties of Determinants - Part 4

If $A$ and $B$ are square matrices of same order:
i) $\operatorname{det}\left(A^{\prime}\right)=\operatorname{det} A$
ii) $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$ and $|A B|=|B A| \mid$
iii) if $A$ is skew symmetric matrix of odd order then $|A|=0$.
iv) if $A$ is a skew symmetric matrix of even order then $|A|$ is a perfect square.
v) $|k A|=k^n|A|$, where $n$ is order of $A$
vii) $\left|A^n\right|=|A|^n$, where $n$ belongs to$N$.