The identity matrix, denoted by I, is a special type of square matrix that has 1s on its main diagonal (from the top left to the bottom right) and 0s elsewhere. When a square matrix is multiplied by its corresponding identity matrix, the result is the original matrix.

Identity matrix of a 2 \times 2 matrix \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The determinant of a square matrix is a scalar value that has several important properties and applications, such as determining the invertibility of a matrix and finding the area of a parallelogram formed by two vectors. The determinant of a 2 \times 2 matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} can be found using the formula \det(A) = |A| =ad -bc.

For example, the determinant of A = \begin{bmatrix} 4 & 2 \\ 3 & 5 \end{bmatrix} would be \det(A) = (4)(5)-(2)(3) = 20 - 6 = 14.

For the matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, a parallelogram can be formed with the vertices (0,\,0),\, (a,\, b)\,(c,\, d), and (a+b,\, c+d). The area of this parallelogram can be found by taking the non-zero absolute value of the determinant. If \det(A) = 0, then the vectors formed by the column and row vectors are parallel.

Example: Compute the area of the parallelogram spanned by the matrix A = \begin{bmatrix} 2 & 6 \\ 6 & 0 \end{bmatrix}. The visual of the parallelogram is represented below.

The area can be found with the formula: |\det(A)| = |ad - bc|.

\displaystyle \|\det(A)\|	\displaystyle =	\displaystyle \|(2)(0) - (6)(6)\|	Substitute a=2,\,d=0,\,b=6 and c=6
	\displaystyle =	\displaystyle \|0-36\|	Evaluate the multiplication
	\displaystyle =	\displaystyle \|-36\|	Evaluate the subtraction
	\displaystyle =	\displaystyle 36	Evaluate

In the next section, we will explore how to use the determinant to determine the inverse of a matrix. A square matrix A has an inverse if and only if its determinant, \det(A) \neq 0.

Examples

Example 1

Find the determinant of the following matrices:

\begin{bmatrix} 5 & 3 \\ 1 & 7 \end{bmatrix}

Worked Solution

Create a strategy

Remember that determinant of a 2 \times 2 matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and use the formula |\det(A)| = |ad - bc| .

Apply the idea

\displaystyle \|\det(A)\|	\displaystyle =	\displaystyle \|(5)(7) - (3)(1)\|	Substitute a = 5,\, b = 3,\, c = 1,\, and d = 7
	\displaystyle =	\displaystyle \|35 - 3\|	Evaluate the product
	\displaystyle =	\displaystyle 32	Evaluate

\begin{bmatrix} 8 & 3 \\ -4 & 2 \end{bmatrix}

Worked Solution

Create a strategy

Remember that determinant of a 2 \times 2 matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and use the formula |\det(A)| =| ad - bc |.

Apply the idea

\displaystyle \|\det(A)\|	\displaystyle =	\displaystyle \|(8)(2) - 3(-4)\|	Substitute a = 8,\, b = 3,\, c = -4,\, and d = 2
	\displaystyle =	\displaystyle 16 + 12	Evaluate the product
	\displaystyle =	\displaystyle 28	Evaluate

Example 2

Determine the area of the parallelogram formed by the column vectors of the matrix A = \begin{bmatrix} 7 & 2 \\ 5 & 9 \end{bmatrix}.

Worked Solution

Create a strategy

For the matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, a parallelogram can be formed with the vertices (0,\,0),\, (a,\, b),\,(c,\, d), and (a+b,\, c+d). The area is the absolute value of the determinant of the matrix.

Apply the idea

The area is the absolute value of the determinant of the matrix.

\displaystyle \|\text{det}(A)\|	\displaystyle =	\displaystyle \|ad - bc\|	Write the formula
	\displaystyle =	\displaystyle \|(7)(9)-(2)(5)\|	Substitute a = 7,\, b = 2,\, c = 5,\, and d = 9
	\displaystyle =	\displaystyle \|63 - 10\|	Evaluate the multiplication
	\displaystyle =	\displaystyle \|53\|	Evaluate the sum
	\displaystyle =	\displaystyle 53	Simplify

The area of the parallelogram is 53.

The visual representation of the parallelogram formed by the column vectors is below:

Idea summary

An identity matrix is a square matrix with 1s on the diagonal from the top left to the bottom right and 0s everywhere else, and it doesn't change a matrix when multiplied together. The determinant of a 2 \times 2 matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} is calculated as ad - bc, and its absolute value can be used to find the area of a parallelogram formed by the column or row vectors of the matrix.

The inverse of a square 2x2 matrix

The inverse of a square matrix, when it exists, is the matrix that, when multiplied by the original matrix, results in the identity matrix \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. This relationship can be described as AA^{-1}=I where A is a 2 \times 2 matrix, A^{-1} is the inverse of the matrix, and I is the identity matrix. For the 2 \times 2 matrix A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, the inverse of the matrix is determined by A^{-1}=\dfrac{1}{\text{det}(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.

A square matrix A is invertible if and only if its determinant, \text{det}(A) \neq 0. If \text{det}(A) = 0, the matrix is called singular or non-invertible.

Examples

Example 3

Determine if the following matrix has an inverse: \begin{bmatrix} 8 & 3 \\ -4 & 2 \end{bmatrix}

Worked Solution

Create a strategy

A square matrix is invertible if and only if \text{det}(A) \neq 0. Find the determinant to see if the matrix is invertible.

Apply the idea

\displaystyle \|\text{det}(A)\|	\displaystyle =	\displaystyle \|ad - bc\|	Write the formula
	\displaystyle =	\displaystyle \|(8)(2) - (3)(-4)\|	Substitute a = 8,\, b = 3,\, c = -4,\, and d = 2
	\displaystyle =	\displaystyle \|16 + 12\|	Evaluate the multiplication
	\displaystyle =	\displaystyle \|28\|	Evaluate the sum
	\displaystyle =	\displaystyle 28	Simplify

The determinant is not equal to zero, so the matrix is invertible.

Example 4

Given the matrix: A = \begin{bmatrix} 9 & -6 \\ -11 & 8 \end{bmatrix}

Determine the inverse of the matrix.

Worked Solution

Create a strategy

We need to first find the determinant and then apply the formula of the inverse of a 2 \times 2 square matrix, A, the inverse can be found as A^{-1} = \dfrac{1}{\text{det}(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.

Apply the idea

\displaystyle \|\text{det}(A)\|	\displaystyle =	\displaystyle \|ad - bc\|	Write the determinant formula
	\displaystyle =	\displaystyle \|(9)(8) - (-6)(-11)\|	Substitute a = 9,\, b = -6,\, c = -11,\, and d = 8
	\displaystyle =	\displaystyle \|72 - 66\|	Evaluate the multiplication
	\displaystyle =	\displaystyle \|6\|	Evaluate the sum
	\displaystyle =	\displaystyle 6	Simplify

Apply the formula of the inverse of a 2 \times 2 matrix.

\displaystyle A^{-1}	\displaystyle =	\displaystyle \dfrac{1}{6}\begin{bmatrix} 8 & 6 \\ 11 & 9 \end{bmatrix}	Substitute a = 9,\, b = -6,\, c = -11,\, and d=8
	\displaystyle =	\displaystyle \begin{bmatrix} 8 \cdot \dfrac{1}{6} & 6 \cdot \dfrac{1}{6} \\ 11 \cdot \dfrac{1}{6} & 9 \cdot \dfrac{1}{6} \end{bmatrix}	Multiply \dfrac{1}{6} in each element
	\displaystyle =	\displaystyle \begin{bmatrix} \dfrac{4}{3} & 1 \\ \dfrac{11}{6} & \dfrac{3}{2} \end{bmatrix}	Evaluate

Find AA^{-1}.

Worked Solution

Create a strategy

Multiply the rows in Matrix A together with the columns of the inverse matrix.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \begin{bmatrix} 9 & -6 \\ -11 & 8 \end{bmatrix}\begin{bmatrix} \dfrac{4}{3} & 1 \\ \dfrac{11}{6} & \dfrac{3}{2} \end{bmatrix}	Write the two matrix
	\displaystyle =	\displaystyle \begin{bmatrix} 9 \cdot \dfrac{4}{3} + (-6) \dfrac{11}{6} & 9 \cdot 1 +(-6) \dfrac{3}{2}\\ (-11)\dfrac{4}{3} + 8 \cdot \dfrac{11}{6} & (-11)\cdot 1 + 8 \cdot \dfrac{3}{2} \end{bmatrix}	Multiply the rows and columns of the two matrix
	\displaystyle =	\displaystyle \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}	Simplify each element

Idea summary

The inverse of a 2\times 2 matrix exists if its determinant is non-zero, and when multiplied by its inverse, a matrix results in the identity matrix. To find the inverse of a 2\times 2 matrix, you can use the formula A^{-1} = \dfrac{1}{\text{det}(A)}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.

4.11 The inverse and determinant of a matrix

Introduction

Ideas

Identity matrix and determinants

Examples

Example 1

Example 2

The inverse of a square 2x2 matrix

Examples

Example 3

Example 4

Outcomes

4.11.A

4.11.B

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