Q.Evaluate the determinant $\begin{vmatrix} 1 & 0 & 2 \\ 3 & 1 & -1 \\ 2 & 4 & 1 \end{vmatrix}$ by expanding along the third column.

Compute each $2\times2$ determinant; only (b) is non-zero, giving $2$.

For a $2\times2$ determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc$.

• (a) $\begin{vmatrix} x & 1 \\ x^2+x & x+1 \end{vmatrix}=x(x+1)-1\cdot(x^2+x)=x^2+x-x^2-x=0$.

• (b) $\begin{vmatrix} x+3 & x+1 \\ x+4 & x+2 \end{vmatrix}=(x+3)(x+2)-(x+1)(x+4)=(x^2+5x+6)-(x^2+5x+4)=2$.

• (c) $\begin{vmatrix} x & 1 \\ 2x & 2 \end{vmatrix}=2x-2x=0$ (second row is $2\times$ first).

• (d) $\begin{vmatrix} 2x^2+2x & x+1 \\ 2x & 1 \end{vmatrix}=(2x^2+2x)-2x(x+1)=2x^2+2x-2x^2-2x=0$.

One-word term: Singular matrix.

For a square matrix $A$, if $|A|=0$ the matrix is singular (it has no inverse). If $|A|\neq 0$ it is non-singular/invertible. This is a standard CHSE Odisha +2 Business Mathematics term.

Correction: "third" should be second order.

The order of a determinant equals the number of its rows (= columns). $\begin{vmatrix} a & d \\ b & c \end{vmatrix}$ is $2\times2$, hence a second order determinant.

Determinant $\neq 0 \Rightarrow$ non-singular matrix.

A square matrix $A$ is classified by the value of its determinant $|A|$:

• If $|A|=0$, the matrix is singular (it has no inverse).
• If $|A|\neq 0$, the matrix is non-singular (it is invertible, since $A^{-1}=\dfrac{1}{|A|}\,\text{adj}(A)$ requires $|A|\neq 0$).

Since the question states the determinant is not equal to zero, the matrix is non-singular.

$\begin{vmatrix} 2 & 3 \\ 0 & 5 \end{vmatrix}=10$.

For a $2\times 2$ determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc$.

Here $a=2,\ b=3,\ c=0,\ d=5$:

$\begin{vmatrix} 2 & 3 \\ 0 & 5 \end{vmatrix}=(2)(5)-(3)(0)=10-0=10$

Vertical bars + $3\times 3$ size ⇒ option (a).

Two things identify a third-order determinant:

1. It must be enclosed in vertical bars $\begin{vmatrix}\;\end{vmatrix}$ — options (b), (c), (d) use square brackets, which denote matrices, not determinants.
2. Its order must be $3$, i.e. a $3\times 3$ array.

Only option (a), $\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}$, satisfies both.

Correct option: (b) 4.

A determinant of order $n$ corresponds to an $n \times n$ arrangement and so contains $n^2$ elements. For order two,

$\begin{vmatrix} a & b \\ c & d \end{vmatrix},$

the number of elements is $2^2 = 4$.