Properties of Matrix Determinant

Determinants of Matrix Products and Scaled Matrices

Suppose we have two matrices, $A$ and $B$ , as follows:

A = \begin{bmatrix} -5 & 2 \\ 1 & -3 \end{bmatrix}

Determinant of the Product of Two Matrices

Investigate the relationship between the determinant of the product of two matrices ( $|AB|$ ) and the product of their individual determinants ( $|A||B|$ ).

Step 1: Calculate the Determinant of Matrix $A$ ( $|A|$ )

\begin{aligned} |A| &= (-5 \cdot -3) - (2 \cdot 1) \\ &= 15 - 2 \\ &= 13 \end{aligned}

Step 2: Calculate the Determinant of Matrix $B$ ( $|B|$ )

\begin{aligned} |B| &= (9 \cdot -1) - (-4 \cdot 1) \\ &= -9 - (-4) \\ &= -9 + 4 \\ &= -5 \end{aligned}

Step 3: Determine the Product of Matrix $A$ and $B$ ( $AB$ )

Matrix $AB$ is obtained by multiplying matrix $A$ and $B$ :

AB = \begin{bmatrix} -5 & 2 \\ 1 & -3 \end{bmatrix} \begin{bmatrix} 9 & -4 \\ 1 & -1 \end{bmatrix}

Compute each entry by multiplying a row by a column:

\begin{aligned} &m_{11}=(-5)(9) + (2)(1) \\ &m_{12}=(-5)(-4) + (2)(-1) \\ &m_{21}=(1)(9) + (-3)(1) \\ &m_{22}=(1)(-4) + (-3)(-1) \end{aligned}

Step 4: Calculate the Determinant of Matrix $AB$ ( $|AB|$ )

Now, calculate the determinant of the product matrix $AB$ :

\begin{aligned} |AB| &= (-43 \cdot -1) - (18 \cdot 6) \\ &= 43 - 108 \\ &= -65 \end{aligned}

Step 5: Compare $|AB|$ with $|A||B|$

We have obtained $|A| = 13$ , $|B| = -5$ , and $|AB| = -65$ .

Calculate $|A||B|$ :

|A||B| = 13 \cdot (-5) = -65

Notice that the value of $|AB|$ is the same as the value of $|A||B|$ .

Formula for the Property of Determinant of Matrix Product

If $A$ and $B$ are two square matrices of the same order, then the determinant of the product of matrices $A$ and $B$ is equal to the product of their individual determinants.

|AB| = |A||B|

Determinant of a Matrix with Scalar Multiplication

Now, investigate what happens to the determinant of a matrix if each element of the matrix is multiplied by a scalar (constant).

Suppose we use matrix $A$ from the previous example and a scalar $k=2$ .

A = \begin{bmatrix} -5 & 2 \\ 1 & -3 \end{bmatrix}

We already know that $|A| = 13$ . Matrix $A$ is a $2 \times 2$ order matrix, so $n=2$ .

Step 1: Determine Matrix $kA$

Multiply each element of matrix $A$ by the scalar $k=2$ :

\begin{aligned} kA &= 2A \\ &= 2\begin{bmatrix} -5 & 2 \\ 1 & -3 \end{bmatrix} \\ &= \begin{bmatrix} 2(-5) & 2(2) \\ 2(1) & 2(-3) \end{bmatrix} \\ &= \begin{bmatrix} -10 & 4 \\ 2 & -6 \end{bmatrix} \end{aligned}

Step 2: Calculate the Determinant of Matrix $kA$ ( $|kA|$ )

\begin{aligned} |kA| &= (-10 \cdot -6) - (4 \cdot 2) \\ &= 60 - 8 \\ &= 52 \end{aligned}

Step 3: Compare $|kA|$ with $k^n|A|$

We have $|kA| = 52$ . The scalar $k=2$ , the order of the matrix $n=2$ , and $|A|=13$ .

Calculate $k^n|A|$ :

k^n|A| = 2^2 \cdot 13 = 4 \cdot 13 = 52

Notice that the value of $|kA|$ is the same as the value of $k^n|A|$ .

Formula for the Property of Determinant of Scalar Multiplication

If $A$ is a square matrix of order $n \times n$ and $k$ is a scalar, then the determinant of matrix $kA$ is $k^n$ multiplied by the determinant of matrix $A$ .

|kA| = k^n |A|

Here, $n$ is the order (number of rows or columns) of the square matrix $A$ .

Other Determinant Properties

The product and whole-matrix scaling laws are part of a larger set of useful rules. In the table, $A$ is square and every row operation produces a new matrix $B$ .

Change or matrix form	Effect on the determinant
Transpose	$\det(A^T)=\det(A)$
Inverse, when $\det(A)\neq0$	$\det(A^{-1})=1/\det(A)$
Triangular matrix $U$	$\det(U)=\prod_{i=1}^{n}u_{ii}$
Swap two rows, $R_i\leftrightarrow R_j$	$\det(B)=-\det(A)$
Multiply one row by $k$	$\det(B)=k\det(A)$
Add a multiple of one row to another, $R_i\leftarrow R_i+kR_j$	$\det(B)=\det(A)$
Two equal rows, or one zero row	$\det(A)=0$

Do not confuse scaling one row with scaling the whole matrix. In an $n\times n$ matrix, $kA$ scales all $n$ rows by $k$ , so the factor $k$ appears $n$ times and gives $k^n$ .

Check Your Understanding

Start with:

A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\qquad \det(A)=-2

The determinant rules predict each result before any full expansion:

B=\begin{bmatrix}3&4\\1&2\end{bmatrix}\quad\Longrightarrow\quad\det(B)=2

Matrix $B$ swaps the rows, matrix $C$ multiplies the first row by $5$ , and matrix $D$ applies $R_2\leftarrow R_2+3R_1$ . Direct calculation confirms all three predictions.