Properties of Matrix Determinant: Matrix - Mathematics (Grade 11)

Discovering Properties of Matrix Determinants

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

Let's 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 $A$ ( $|A|$ )

The determinant of matrix $A$ is:

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

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

The determinant of matrix $B$ is:

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

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}

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

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

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

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

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

Let's 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, 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 $k=2$ .

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 $k=2$ :

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 $kA$ ( $|kA|$ )

The determinant of matrix $kA$ is:

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

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

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

Let's 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 $A$ .

Discovering Properties of Matrix Determinants

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

Let's 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 $A$ ( $|A|$ )

The determinant of matrix $A$ is:

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

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

The determinant of matrix $B$ is:

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

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}

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

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

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

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

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

Let's 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, 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 $k=2$ .

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 $k=2$ :

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 $kA$ ( $|kA|$ )

The determinant of matrix $kA$ is:

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

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

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

Let's 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 $A$ .