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Explore matrix determinant properties: product rule |AB| = |A||B| and scalar multiplication |kA| = k^n|A|. Learn through worked examples.

---

## Discovering Properties of Matrix Determinants

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

Visible text: Suppose we have two matrices, and , as follows:

Component: MathContainer
Children:

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

```math
B = \begin{bmatrix} 9 & -4 \\ 1 & -1 \end{bmatrix}
```

### Determinant of the Product of Two Matrices

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

Visible text: Let's investigate the relationship between the determinant of the product of two matrices () and the product of their individual determinants ().

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

Visible text: **Step : Calculate the Determinant of Matrix ()**

The determinant of matrix $$A$$ is:

Visible text: The determinant of matrix is:

```math
|A| = (-5 \cdot -3) - (2 \cdot 1) = 15 - 2 = 13
```

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

Visible text: **Step : Calculate the Determinant of Matrix ()**

The determinant of matrix $$B$$ is:

Visible text: The determinant of matrix is:

```math
|B| = (9 \cdot -1) - (-4 \cdot 1) = -9 - (-4) = -9 + 4 = -5
```

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

Visible text: **Step : Determine the Product of Matrix and ()**

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

Visible text: Matrix is obtained by multiplying matrix and :

Component: MathContainer
Children:

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

```math
AB = \begin{bmatrix} (-5)(9) + (2)(1) & (-5)(-4) + (2)(-1) \\ (1)(9) + (-3)(1) & (1)(-4) + (-3)(-1) \end{bmatrix}
```

```math
AB = \begin{bmatrix} -45 + 2 & 20 - 2 \\ 9 - 3 & -4 + 3 \end{bmatrix}
```

```math
AB = \begin{bmatrix} -43 & 18 \\ 6 & -1 \end{bmatrix}
```

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

Visible text: **Step : Calculate the Determinant of Matrix ()**

Now, let's calculate the determinant of the product matrix $$AB$$:

Visible text: Now, let's calculate the determinant of the product matrix :

```math
|AB| = (-43 \cdot -1) - (18 \cdot 6) = 43 - 108 = -65
```

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

Visible text: **Step : Compare with **

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

Visible text: We have obtained , , and .

Let's calculate $$|A||B|$$:

Visible text: Let's calculate :

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

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

Visible text: Notice that the value of is the same as the value of .

**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.

Visible text: If and are two square matrices of the same order, then the determinant of the product of matrices and is equal to the product of their individual determinants.

```math
|AB| = |A||B|
```

### Determinant of a Matrix with Scalar Multiplication

Now, let's 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 $$k=2$$.

Visible text: Suppose we use matrix from the previous example and a scalar .

```math
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 $$n=2$$.

Visible text: We already know that . Matrix is a order matrix, so .

**Step $$1$$: Determine Matrix $$kA$$**

Visible text: **Step : Determine Matrix **

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

Visible text: Multiply each element of matrix by the scalar :

```math
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}
```

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

Visible text: **Step : Calculate the Determinant of Matrix ()**

The determinant of matrix $$kA$$ is:

Visible text: The determinant of matrix is:

```math
|kA| = (-10 \cdot -6) - (4 \cdot 2) = 60 - 8 = 52
```

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

Visible text: **Step : Compare with **

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

Visible text: We have . The scalar , the order of the matrix , and .

Let's calculate $$k^n|A|$$:

Visible text: Let's calculate :

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

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

Visible text: Notice that the value of is the same as the value of .

**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 $$A$$.

Visible text: If is a square matrix of order and is a scalar, then the determinant of matrix is multiplied by the determinant of matrix .

```math
|kA| = k^n |A|
```

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

Visible text: Here, is the order (number of rows or columns) of the square matrix .