Q.If $A$ is a matrix of order $3 \times 3$, then $|3A| =$ ________ .

The key idea is scalar multiplication in determinants: when a matrix is multiplied by a scalar, each row (or column) is multiplied by that scalar.

For a $3 \times 3$ matrix $A$, the determinant of $3A$ is found by factoring out the scalar from each of the 3 rows: …

The key idea here is how the determinant behaves when you multiply a matrix by a constant. Many students rush and think $|3A| = 3|A|$, but that’s only true for a $1 \times 1$ matrix. For larger matrices, the scalar multiplies every row, and the determinant picks up a factor from each row.

Think of it this way: if you take a $3 \times 3$ matrix $A$ and multiply it by $3$, you are multiplying each of its three rows by $3$. The determinant is a multilinear function in the rows — meaning if you multiply a single row by $3$, the determinant gets multiplied by $3$. Multiply all three rows by $3$, and you multiply the determinant by $3$ three times, i.e., $3^3 = 27$.

Let’s walk through it step by step.

1. Recall the property for a single row scaling.

   If $B$ is the matrix obtained from $A$ by multiplying one row by a scalar $k$, then $|B| = k\,|A|$. This is a fundamental property of determinants.

2. Apply it to all rows.

   $3A$ means every entry of $A$ is multiplied by $3$. Equivalently, each of the three rows of $A$ is multiplied by $3$. So we can think of building $3A$ from $A$ in three steps: multiply row 1 by 3, then row 2 by 3, then row 3 by 3.

3. Track the determinant after each step.

   • After scaling row 1: determinant becomes $3|A|$.
   • Then scale row 2: determinant becomes $3 \cdot (3|A|) = 3^2 |A|$. …

Method: Finding How a Scalar Multiplying a Whole Matrix Affects Its Determinant

Whenever a question asks for $|kA|$ in terms of $|A|$, remember that multiplying a matrix by a scalar multiplies every row by that scalar — and a determinant picks up one factor of $k$ for each row it scales, not just one overall factor of $k$.

Steps

Step 1: Recall the single-row scaling property

If a matrix $B$ is obtained from $A$ by multiplying just one row by $k$, then $|B|=k\,|A|$ — this is the multilinear-in-rows property of determinants, and it is the building block for everything else in this method.

Step 2: Recognise that $kA$ scales every row, one at a time

$kA$ multiplies all $n$ rows of an $n\times n$ matrix $A$ by $k$ simultaneously. Mentally build $kA$ from $A$ by scaling row 1, then row 2, ..., then row $n$ — applying Step 1's rule once per row.

Step 3: Multiply the per-row factors together …

Common Mistakes

Mistake 1: Writing $|kA| = k\,|A|$, forgetting the exponent

Why it's wrong: this treats the determinant as if only one row (or the matrix as a single number) were being scaled, but $kA$ scales every one of the $n$ rows, so the determinant is multiplied by $k$ a total of $n$ times, giving $k^n$, not $k$. This is only correct for a $1\times1$ matrix. Correct approach: always ask "how many rows does the scalar actually touch?" — for an $n\times n$ matrix, that answer is $n$, so the determinant scales by $k^n$.

Mistake 2: Using the wrong order $n$ when substituting into $k^n$ …