Key Concepts:
Matrix Definition:
A matrix is an array of numbers, symbols, or expressions arranged in rows and columns. For example: \[ A = \begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \end{bmatrix} \]Size of a Matrix:
The size of a matrix is defined as the number of rows \(\times\) the number of columns. For example, the above matrix \(A\) is a \(3 \times 3\) matrix.Basic Operations with Matrices:
Addition and Subtraction:
Matrices of the same size are added or subtracted element by element.Multiplication:
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. \\ The element in the \(i\)th row and \(j\)th column of the product matrix is the sum of the products of the corresponding elements of the \(i\)th row of the first matrix and the \(j\)th column of the second matrix. \\ \[ \begin{bmatrix} a_1 & b_1 \\\\ c_1 & d_1 \end{bmatrix} \cdot \begin{bmatrix} a_2 & b_2 \\\\ c_2 & d_2 \end{bmatrix} = \begin{bmatrix} a_1a_2 + b_1c_2 & a_1b_2 + b_1d_2 \\\\ c_1a_2 + d_1c_2 & c_1b_2 + d_1d_2 \end{bmatrix} \]Determinant:
The determinant is a scalar value associated with a square matrix, used to determine whether the matrix is invertible. For a \(2 \times 2\) matrix: \[ \text{If } A = \begin{bmatrix} a & b \\\\ c & d \end{bmatrix}, \quad \text{then } \det(A) = ad - bc. \]Inverse Matrix:
The inverse of a square matrix \(A\), denoted \(A^{-1}\), exists if and only if \(\det(A) \neq 0\). For a \(2 \times 2\) matrix: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\\\ -c & a \end{bmatrix}. \]Example 1: Add the matrices \(A = \begin{bmatrix} 1 & 2 \\\\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 6 \\\\ 7 & 8 \end{bmatrix}\).
Solution:
Add corresponding elements: \[ A + B = \begin{bmatrix} 1+5 & 2+6 \\\\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\\\ 10 & 12 \end{bmatrix}. \] Therefore, \(A + B = \begin{bmatrix} 6 & 8 \\\\ 10 & 12 \end{bmatrix}\).Example 2: Find the determinant of \(A = \begin{bmatrix} 3 & 4 \\\\ 2 & 1 \end{bmatrix}\).
Solution:
Use the formula for the determinant of a \(2 \times 2\) matrix: \[ \det(A) = (3)(1) - (4)(2) = 3 - 8 = -5. \] Therefore, \(\det(A) = -5\).Example 3: Multiply the matrices \(A = \begin{bmatrix} 1 & 2 \\\\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 0 \\\\ 1 & 3 \end{bmatrix}\).
Solution:
Use the rule for matrix multiplication: \[ A \cdot B = \begin{bmatrix} (1)(2)+(2)(1) & (1)(0)+(2)(3) \\\\ (3)(2)+(4)(1) & (3)(0)+(4)(3) \end{bmatrix} = \begin{bmatrix} 4 & 6 \\\\ 10 & 12 \end{bmatrix}. \] Therefore, \(A \cdot B = \begin{bmatrix} 4 & 6 \\\\ 10 & 12 \end{bmatrix}\).Example 4: Find the inverse of \(A = \begin{bmatrix} 2 & 3 \\\\ 1 & 4 \end{bmatrix}\).
Solution:
Step 1: Calculate the determinant: \[ \det(A) = (2)(4) - (3)(1) = 8 - 3 = 5. \] Step 2: Use the formula for the inverse of a \(2 \times 2\) matrix: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\\\ -c & a \end{bmatrix} = \frac{1}{5} \begin{bmatrix} 4 & -3 \\\\ -1 & 2 \end{bmatrix}. \] Therefore: \[ A^{-1} = \begin{bmatrix} \frac{4}{5} & -\frac{3}{5} \\\\ -\frac{1}{5} & \frac{2}{5} \end{bmatrix}. \]