Matrices (addition, multiplication, determinants) — KCSE Mathematics

To multiply two matrices, ensure they are compatible. A matrix of size m×n can be multiplied by a matrix of size n×p. The result will be a new matrix of size m×p.

Key points for matrix multiplication:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Matrix multiplication is not commutative; that is, AB ≠ BA in general.

Example: Let matrix A be: [ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ] Let matrix B be: [ B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} ]

To multiply A and B:

1. Confirm compatibility: A is 2×2 and B is 2×2, so multiplication is defined.
2. Calculate: [ AB = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix} ]

To calculate the determinant of a 2×2 matrix, use the formula:
For a matrix ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ), the determinant ( |A| ) is given by:
[ |A| = ad - bc ]
The inverse of a 2×2 matrix exists if the determinant is not equal to zero (det ≠ 0). The formula for the inverse is:
[ A^{-1} = \frac{1}{|A|} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} ]
Example:
If ( A = \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix} ):

1. Calculate the determinant:
   • ( |A| = (2)(4) - (3)(1) = 8 - 3 = 5 )
2. Since ( |A| \neq 0 ), the inverse exists:
   • ( A^{-1} = \frac{1}{5} \begin{pmatrix} 4 & -3 \ -1 & 2 \end{pmatrix} )
     Thus, the inverse is ( A^{-1} = \begin{pmatrix} \frac{4}{5} & -\frac{3}{5} \ -\frac{1}{5} & \frac{2}{5} \end{pmatrix} ).

Matrices are rectangular arrays of numbers that can represent data or solve systems of equations. Key operations include:

• Addition: Two matrices can be added if they have the same dimensions. The sum is obtained by adding corresponding elements.
• Multiplication: To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. The resulting matrix's element is the sum of the products of corresponding elements from the row of the first matrix and the column of the second.
• Determinants: The determinant is a scalar value that can be computed from a square matrix. It provides important properties about the matrix, such as whether it is invertible.

Example of Matrix Addition: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then: A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].

Example of Matrix Multiplication: If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then: A × B = [[(1×5 + 2×7), (1×6 + 2×8)], [(3×5 + 4×7), (3×6 + 4×8)]] = [[19, 22], [43, 50]].