Matrices and Determinants - Examples & Applications

This section provides step-by-step examples of matrix operations. You will learn to add, subtract, and multiply matrices, calculate determinants, find the inverse of a 2×22 \times 2 matrix, and apply Cramer's rule to solve systems of equations.

Matrix Operations

Matrix operations have strict rules regarding dimensions. Addition and subtraction require identical dimensions, while multiplication requires the inner dimensions to match.

Example

Example 1: Scalar Multiplication and Addition (Basic) Given matrices AA and BB, find 3A−2B3A - 2B: A=[2−104]A = \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} B=[52−31]B = \begin{bmatrix} 5 & 2 \\ -3 & 1 \end{bmatrix}

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Example

Example 2: Matrix Multiplication (Intermediate) Find the product ABAB: A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} (a 2×22 \times 2 matrix) B=[−10521−3]B = \begin{bmatrix} -1 & 0 & 5 \\ 2 & 1 & -3 \end{bmatrix} (a 2×32 \times 3 matrix)

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Example

Example 3: Non-Commutative Multiplication (Edge Case) Using the matrices from Example 2, what is the product BABA?

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Determinants

The determinant is a scalar value calculated from a square matrix. It has many applications, including finding inverses and solving systems of equations.

Example

Example 1: Determinant of a 2x2 Matrix (Basic) Calculate the determinant of matrix A=[4−235]A = \begin{bmatrix} 4 & -2 \\ 3 & 5 \end{bmatrix}.

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Example

Example 2: Inverse of a 2x2 Matrix (Intermediate) Find the inverse of matrix B=[2153]B = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}.

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Example

Example 3: Determinant of a 3x3 Matrix (Advanced) Calculate the determinant of matrix C=[12−13042−25]C = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 0 & 4 \\ 2 & -2 & 5 \end{bmatrix} using cofactor expansion along the first row.

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Inverse of a Matrix

The inverse of a matrix AA, denoted A−1A^{-1}, satisfies AA−1=IAA^{-1} = I, where II is the identity matrix.

Example

Example 1: Inverse of a 2x2 Matrix (Intermediate) Find the inverse of the matrix A=[4726]A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}.

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Cramer's Rule

Cramer's rule provides an explicit formula for the solution of a system of linear equations using determinants.

Example

Case Study 1: Solving a 2x2 System with Cramer's Rule Use Cramer's rule to solve the system: 3x−2y=43x - 2y = 4 5x+y=115x + y = 11

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