Q.Find the value of the following:

(i) This is the $3\times3$ identity matrix; a diagonal determinant is the product of the diagonal entries: $1\cdot1\cdot1 = 1.$

(ii) Expand along the first row of $\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}$ (its middle entry is $0$):

A $3\times3$ determinant can be expanded along any row or column, using the sign checkerboard $\begin{pmatrix}+&-&+\\-&+&-\\+&-&+\end{pmatrix}$. Choosing a row or column that contains zeros saves work.

(i)

The matrix is the identity: $1$'s on the diagonal and $0$'s everywhere else. A diagonal (in fact triangular) determinant is the product of the diagonal entries, so the value is $1\cdot1\cdot1 = 1.$

(ii)

Expand along row 1; the $0$ in the middle kills that term:

The minors are $\begin{vmatrix} 5 & -1 \\ 1 & 2 \end{vmatrix} = 10-(-1) = 11$ and $\begin{vmatrix} 3 & 5 \\ 0 & 1 \end{vmatrix} = 3-0 = 3.$

So the value is $1(11) + 4(3) = 11+12 = 23.$

Method: Recognising Special Matrix Structure Before Expanding

A time-saving check to run before committing to a full cofactor expansion.

Steps

Step 1: Check for special structure first

• Identity or diagonal matrix: determinant is the product of the diagonal entries (instantly).
• Triangular matrix: same shortcut — product of the diagonal entries.

Step 2: If no shortcut applies, choose the row/column with the most zeros

Fewer nonzero entries means fewer cofactor terms to compute.

Step 3: Expand using cofactors, skipping zero entries entirely

where any term with $a_{ij}=0$ contributes nothing and can be omitted from the sum without computing its minor.

Step 4: Compute the remaining $2\times2$ minors and assemble the answer

Add up the nonzero contributions, tracking the $(-1)^{i+j}$ sign for each.

Common Mistakes

Mistake 1: Expanding the identity matrix's determinant the "long way" via full cofactor expansion

Why it's wrong: this wastes time and adds unnecessary arithmetic when the identity (or any diagonal) matrix's determinant is immediately $1$ by the product-of-diagonal shortcut. Correct approach: check for identity/diagonal/triangular structure first and read the determinant off instantly when it applies.

Mistake 2: Still computing the $2\times2$ minor for a term whose coefficient is $0$

Why it's wrong: multiplying a computed minor by $0$ always gives $0$, so working out that minor is wasted effort that only increases the chance of an unrelated arithmetic slip elsewhere. Correct approach: when expanding along a row/column with a zero entry, skip that term's minor entirely and move straight to the nonzero terms.