Q.If $A$ and $B$ are matrices of order $3$ and $|A| = 5$, $|B| = 3$, then $|3AB| = 27 \times 5 \times 3 = 405$.

Concept: Scalar Multiplication — When a matrix is multiplied by a scalar $k$, each of its rows is multiplied by $k$, so the determinant gets multiplied by $k^n$ for an $n \times n$ matrix.

Step 1: For a $3 \times 3$ matrix, $|kA| = k^3 |A|$. Here $k = 3$, so $|3A| = 3^3 \cdot 5 = 27 \cdot 5$. …

The key idea is that when you multiply a matrix by a scalar, every single entry gets scaled. But the determinant is not linear in the matrix — it’s multilinear in the rows (or columns). So if you multiply the whole matrix by $3$, each of the $3$ rows gets a factor of $3$, and the determinant picks up a factor of $3^3 = 27$.

Now, $3AB$ means the scalar $3$ multiplies the product $AB$, not $A$ or $B$ individually. So we first combine $A$ and $B$ using the determinant product property, then apply the scalar rule.

1. Determinant of a product For square matrices of the same order, $|AB| = |A| \cdot |B|$. Here $|A| = 5$ and $|B| = 3$, so

2. Effect of scalar multiplication If $C$ is an $n \times n$ matrix, then $|kC| = k^n |C|$. Why? Because multiplying the whole matrix by $k$ multiplies each of the $n$ rows by $k$, and the determinant is multilinear in rows — each row factor comes out separately. Here $n = 3$ and $k = 3$, so …

Method: Computing the Determinant of a Scalar Multiple of a Matrix Product

Use this method whenever you need $|kAB|$ (or similar), for a scalar $k$ and matrices $A, B$ of order $n$ — combine the scalar rule and the product rule in the right order.

Steps

Step 1: Recall the two determinant rules you need

The scalar rule needs the exponent $n$ — NOT just $k$ — because scaling the whole matrix scales every one of its $n$ rows by $k$.

Step 2: Decide what the scalar is multiplying …

Common Mistakes

Mistake 1: Using $|kM| = k|M|$ instead of $|kM| = k^{n}|M|$

Why it's wrong: this is by far the most common slip — students treat scalar multiplication of a determinant like scalar multiplication of a single number, forgetting that scaling a matrix scales EVERY row, so the factor $k$ comes out once per row (i.e. $k^n$, not $k$). Correct approach: always identify the order $n$ of the matrix first, then raise the scalar to that power.

Mistake 2: Splitting the scalar's power across the two matrices in a product …