Grade 12

Advanced

9 min

Lesson 9 of 12

Determinants and Matrix Inversion

Not Started

Calculate determinants and find inverse matrices to solve complex systems of linear equations.

Ever wondered how a GPS calculates your exact position from four different satellites in milliseconds? It is not magic—it is the power of 'unlocking' matrices to solve hidden equations simultaneously.

Learning Objectives

1.Calculate the determinant of $2 \times 2$ and $3 \times 3$ matrices using expansion by cofactors
2.Determine if a matrix is singular or invertible based on its determinant value
3.Find the inverse of a matrix and use it to solve matrix equations of the form $AX = B$

The Determinant: A Matrix's DNA

The determinant is a scalar value that reveals fundamental properties of a square matrix. Think of it as the 'scaling factor' of a linear transformation. For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , the determinant is calculated as $\det(A) = ad - bc$ . If the determinant is zero, the matrix is called singular, meaning it 'collapses' space into a lower dimension and cannot be reversed. If $\det(A) \neq 0$ , the matrix is invertible (non-singular), and a unique inverse exists.

Find the determinant of $M = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$ .

1. Identify the elements: $a=4, b=7, c=2, d=6$ . 2. Apply the formula: $\det(M) = (4)(6) - (7)(2)$ . 3. Calculate: $24 - 14 = 10$ . 4. Since $10 \neq 0$ , the matrix $M$ is invertible.

Quick Check

If a matrix has a determinant of 0, what does this tell us about its inverse?

Think about what happens when you try to divide by the determinant.

Answer

The matrix is singular and does not have an inverse.

Expansion by Cofactors

To find the determinant of a

3 \times 3

matrix, we use Laplace Expansion (expansion by cofactors). This involves breaking the large matrix into smaller

2 \times 2

'minors.' We pick a row or column (usually the one with the most zeros) and multiply each element by its cofactor. The sign of each term follows a 'checkerboard' pattern:

\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}

The determinant is the sum of these products.

Calculate $\det(A)$ for $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{pmatrix}$ expanding along the first column.

1. Term 1: $+1 \cdot \det\begin{pmatrix} 4 & 5 \\ 0 & 6 \end{pmatrix} = 1(24 - 0) = 24$ . 2. Term 2: $-0 \cdot \det\begin{pmatrix} 2 & 3 \\ 0 & 6 \end{pmatrix} = 0$ . 3. Term 3: $+1 \cdot \det\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} = 1(10 - 12) = -2$ . 4. Final sum: $24 - 0 - 2 = 22$ .

Quick Check

Which row or column is usually the most efficient to choose for expansion?

Multiplying by zero cancels out the entire cofactor term.

Answer

The row or column containing the most zeros, as it simplifies the calculations.

Matrix Inversion and Systems

The inverse matrix $A^{-1}$ is the matrix such that $AA^{-1} = I$ , where $I$ is the identity matrix. For a $2 \times 2$ matrix, $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ . This is incredibly powerful for solving systems of linear equations. If we have the equation $AX = B$ , where $A$ is a matrix of coefficients and $B$ is a constant vector, we can find the solution $X$ by multiplying both sides by the inverse: $X = A^{-1}B$ .

Solve the system: $3x + 4y = 10$ $1x + 2y = 4$

1. Write as $AX = B$ : $\begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 10 \\ 4 \end{pmatrix}$ . 2. Find $\det(A) = (3)(2) - (4)(1) = 2$ . 3. Find $A^{-1} = \frac{1}{2} \begin{pmatrix} 2 & -4 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 1 & -2 \\ -0.5 & 1.5 \end{pmatrix}$ . 4. Solve $X = A^{-1}B$ : $\begin{pmatrix} 1 & -2 \\ -0.5 & 1.5 \end{pmatrix} \begin{pmatrix} 10 \\ 4 \end{pmatrix} = \begin{pmatrix} (10-8) \\ (-5+6) \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ . 5. Solution: $x=2, y=1$ .

Key Takeaways

1The determinant $\det(A)$ must be non-zero for an inverse matrix to exist.
2For $2 \times 2$ matrices, $\det(A) = ad - bc$ .
3Expansion by cofactors reduces a $3 \times 3$ determinant into three $2 \times 2$ determinants.
4Matrix equations $AX = B$ are solved using the formula $X = A^{-1}B$ .

Quick Check

Multiple Choice

What is the determinant of $\begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix}$ ?

Multiple Choice

A matrix $A$ is singular if:

True/False

To solve $AX = B$ , you can calculate $X = B A^{-1}$ .

What's Next?

Review Tomorrow

In 24 hours, try to write down the $2 \times 2$ inverse formula from memory and explain why the determinant cannot be zero.

Practice Activity

Create a $3 \times 3$ matrix with at least two zeros and calculate its determinant using expansion by cofactors along the row with those zeros.

Browse

Browse

Determinants and Matrix Inversion

Learning Objectives

The Determinant: A Matrix's DNA

Expansion by Cofactors

Matrix Inversion and Systems

Key Takeaways

Quick Check

What's Next?

Determinants and Matrix Inversion

Learning Objectives

The Determinant: A Matrix's DNA

Expansion by Cofactors

Matrix Inversion and Systems

Key Takeaways

Quick Check

What's Next?