Question

Let $\mathbf{X}=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix}$ be a $3\times 3$ matrix. The determinant of $\mathbf{X}$ is $5$. The determinant of matrix $\mathbf{Y}=\begin{bmatrix} x_{11} & x_{12} & x_{13} \\ 2x_{21} & 2x_{22} & 2x_{23} \\ 3x_{31} & 3x_{32} & 3x_{33} \end{bmatrix}$ is __________.

Hint:

Row scaling rule: multiplying $k$-th row by $c$ scales $\det$ by $c$. Multiple rows scaled \Rightarrow multiply all the scalars.

Answer 1

Step 1: Use determinant scaling by rows.
Multiplying a single row by a scalar $c$ multiplies the determinant by $c$.
Step 2: Apply to $\mathbf{Y$.}
Row 1 is unchanged $(\times 1)$, Row 2 is multiplied by $2$, Row 3 is multiplied by $3$. Hence,
\[ \det(\mathbf{Y})=(1)\cdot(2)\cdot(3)\,\det(\mathbf{X})=6\times 5=30. \]