Matrices and Determinants

Edwin Ogie Library

Matrices and Determinants

Objectives:
- Perform basic operations (×, +, -, ÷) on matrices (up to 3×3).
- Calculate determinants of matrices (up to 3×3).
- Compute inverses of 2×2 matrices.

Introduction to Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. For example, a 2×2 matrix is written as:

⎡ 1 2 ⎤
⎣ 3 4 ⎦

Matrices are used in many areas of mathematics and applied sciences to represent data, solve systems of linear equations, and perform various transformations.

Basic Operations on Matrices

Addition/Subtraction: Two matrices of the same dimensions are added or subtracted by performing the operation elementwise.

Scalar Multiplication: Multiply every element of the matrix by a constant.

Matrix Multiplication: For an m×n matrix A and an n×p matrix B, the product AB is an m×p matrix where each element is the dot product of the corresponding row of A and column of B.

For example, if A = ⎡ 1 2 ⎤ and B = ⎡ 5 6 ⎤, then
            ⎣ 3 4 ⎦        ⎣ 7 8 ⎦
A × B (for 2×2 multiplication) is computed as shown in the worked examples.

Determinants

The determinant of a matrix is a scalar value that provides important properties such as invertibility.

For a 2×2 matrix A = ⎡ a b ⎤
          ⎣ c d ⎦, the determinant is:

det(A) = ad - bc

For a 3×3 matrix A = ⎡ a b c ⎤
              ⎢ d e f ⎥
              ⎣ g h i ⎦, the determinant is:

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Inverse of a 2×2 Matrix

A 2×2 matrix A = ⎡ a b ⎤
          ⎣ c d ⎦ has an inverse if and only if det(A) ≠ 0. The inverse is given by:

A^-1 = 1/(ad - bc) ⎡ d  -b ⎤
                  ⎣ -c  a ⎦

Algebra of Matrices (Up to 3×3)

Matrices obey several algebraic properties:

• Associativity: (AB)C = A(BC)
• Distributivity: A(B + C) = AB + AC
• Non-Commutativity: In general, AB ≠ BA

Worked Examples: Matrices and Determinants

Below are 15 worked examples demonstrating basic operations, determinant calculations, and computing inverses.

Summary and Key Concepts

• Matrices: Rectangular arrays representing data.
• Basic Operations: Addition, subtraction, scalar multiplication, and matrix multiplication.
• Determinants: Scalar values that indicate properties like invertibility.
• Inverse of 2×2 Matrices: Computed when the determinant is nonzero.

Mastery of these concepts is essential in linear algebra and its applications.

Extended Discussion and Applications

Matrices are used in solving systems of linear equations, computer graphics, engineering, and more. Determinants help us understand properties such as area, volume scaling, and the invertibility of linear transformations. Inverses are crucial when solving equations of the form AX = B.

15 Worked Examples (Solutions Hidden)

Example 1: Matrix Addition

Question: Add A = ⎡ 1 2 ⎤ and B = ⎡ 3 4 ⎤, where A and B are 2×2 matrices.
A = ⎣ 5 6 ⎦ and B = ⎣ 7 8 ⎦.

Solution:

A + B = ⎡ 1+3 2+4 ⎤ = ⎡ 4 6 ⎤
             ⎣ 5+7 6+8 ⎦ = ⎣ 12 14 ⎦.

Example 2: Matrix Subtraction

Question: Subtract B = ⎡ 3 4 ⎤ from A = ⎡ 9 8 ⎤, where A = ⎣ 7 6 ⎦ and B = ⎣ 1 2 ⎦.

Solution:

A - B = ⎡ 9-3 8-4 ⎤ = ⎡ 6 4 ⎤
             ⎣ 7-1 6-2 ⎦ = ⎣ 6 4 ⎦.

Example 3: Scalar Multiplication

Question: Multiply matrix A = ⎡ 2 3 ⎤ by 4, where A = ⎣ 4 5 ⎦.

Solution:

4A = ⎡ 4×2 4×3 ⎤ = ⎡ 8 12 ⎤
             ⎣ 4×4 4×5 ⎦ = ⎣ 16 20 ⎦.

Example 4: Matrix Multiplication (2×2)

Question: Multiply A = ⎡ 1 2 ⎤ by B = ⎡ 3 4 ⎤, where A = ⎣ 5 6 ⎦ and B = ⎣ 7 8 ⎦.

Solution:

AB = ⎡ (1×3 + 2×7) (1×4 + 2×8) ⎤ = ⎡ (3+14) (4+16) ⎤ = ⎡ 17 20 ⎤
             ⎣ (5×3 + 6×7) (5×4 + 6×8) ⎦ = ⎣ (15+42) (20+48) ⎦ = ⎣ 57 68 ⎦.

Example 5: Determinant of a 2×2 Matrix

Question: Compute det(A) for A = ⎡ 3 5 ⎤, where A = ⎣ 7 9 ⎦.

Solution:

det(A) = (3×9) - (5×7) = 27 - 35 = -8.

Example 6: Determinant of a 3×3 Matrix

Question: Compute det(A) for A = ⎡ 1 2 3 ⎤,
            ⎢ 0 1 4 ⎥,
            ⎣ 5 6 0 ⎦.

Solution:

det(A) = 1[(1×0) - (4×6)] - 2[(0×0) - (4×5)] + 3[(0×6) - (1×5)] = 1(0-24) - 2(0-20) + 3(0-5)
= -24 + 40 - 15 = 1.

Example 7: Inverse of a 2×2 Matrix

Question: Find A^-1 for A = ⎡ 4 7 ⎤, where A = ⎣ 2 6 ⎦.

Solution:

det(A) = (4×6) - (7×2) = 24 - 14 = 10.
A^-1 = (1/10) ⎡ 6  -7 ⎤
                     ⎣ -2  4 ⎦.

Example 8: Matrix Multiplication (3×3)

Question: Multiply A = ⎡ 1 0 2 ⎤,
            ⎢ -1 3 1 ⎥,
            ⎣ 3 2 0 ⎦ by B = ⎡ 2 1 0 ⎤,
                  ⎢ 1 0 1 ⎥,
                  ⎣ 0 2 3 ⎦.

Solution:

Compute each element of the product using the dot product of rows of A and columns of B (detailed calculations provided in class).

Example 9: Scalar Division

Question: Divide A = ⎡ 6 8 ⎤ by 2, where A = ⎣ 10 12 ⎦.

Solution:

A/2 = ⎡ 6/2 8/2 ⎤ = ⎡ 3 4 ⎤
            ⎣ 10/2 12/2 ⎦ = ⎣ 5 6 ⎦.

Example 10: Matrix Subtraction (3×3)

Question: Compute A - B, where
A = ⎡ 3 5 7 ⎤,
             ⎢ 2 4 6 ⎥,
             ⎣ 1 3 5 ⎦ and B = ⎡ 1 2 3 ⎤,
                     B = ⎣ 0 1 2 ⎦,
                         ⎣ 1 1 1 ⎦.

Solution:

A - B = ⎡ (3-1) (5-2) (7-3) ⎤ = ⎡ 2 3 4 ⎤
                         ⎣ (2-0) (4-1) (6-2) ⎦ = ⎣ 2 3 4 ⎦
                         ⎣ (1-1) (3-1) (5-1) ⎦ = ⎣ 0 2 4 ⎦.

Example 11: Verifying Associativity

Question: Verify that matrix addition is associative for matrices A, B, and C of the same dimensions.

Solution:

(A + B) + C = A + (B + C) because addition is done elementwise and the addition of real numbers is associative.

Example 12: Verifying Commutativity

Question: Show that A + B = B + A for two matrices A and B of the same dimensions.

Solution:

Since addition is performed elementwise and addition in ℝ is commutative, A + B equals B + A.

Example 13: Scalar Multiplication (3×3)

Question: Multiply A = ⎡ 1 2 3 ⎤ by 2, where A = ⎣ 4 5 6 ⎦, and A = ⎣ 7 8 9 ⎦.

Solution:

2A = ⎡ 2 4 6 ⎤
            ⎢ 8 10 12 ⎥
            ⎣ 14 16 18 ⎦.

Example 14: Computing Determinant (2×2)

Question: Find the determinant of A = ⎡ 5 3 ⎤, where A = ⎣ 2 4 ⎦.

Solution:

det(A) = (5×4) - (3×2) = 20 - 6 = 14.

Example 15: Inverse of a 2×2 Matrix

Question: Compute the inverse of A = ⎡ 2 3 ⎤, where A = ⎣ 1 4 ⎦.

Solution:

det(A) = (2×4) - (3×1) = 8 - 3 = 5.
A^-1 = (1/5) ⎡ 4  -3 ⎤
                     ⎣ -1  2 ⎦.

30 CBT JAMB Quiz on Matrices and Determinants

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