Matrices and Systems of Linear Equations

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1Summary 重點摘要

Matrices arrange the coefficients of linear systems into tables: master addition, scalar multiplication, multiplication and determinants, solve systems of two and three linear equations by inverse matrices, Cramer's rule or Gaussian elimination, and read the existence of solutions from whether the determinant is zero.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Matrices and their basic operationsA matrix arranges numbers into a rectangular table with m rows and n columns, called an m-by-n matrix; equal row and column counts make a square matrix. Equality needs the same order and every entry equal; addition and subtraction need the same order and work entry by entry; scalar multiplication multiplies every entry. For example, a chain store's stock of two products across three branches is a 2-by-3 matrix: merging two stock tables adds them entry by entry, and raising all prices by ten per cent multiplies the matrix by 1.1. Simple rules — but checking the order is always step one.
2. Step 2: Matrix multiplication and the identity matrixMultiplication requires the left matrix's column count to equal the right's row count; the entry in row i, column j of the product is the sum of products pairing row i of the left with column j of the right. For example, A = [1 2; 3 4] times B = [5 6; 7 8]: the top-left entry is 1×5 + 2×7 = 19, and so on for all four entries. Multiplication is associative and distributive but in general AB ≠ BA — never swap the order. The identity matrix I has ones on the diagonal and zeros elsewhere, leaving any matrix unchanged; transposition swaps rows and columns, with (AB)ᵀ = BᵀAᵀ.
3. Step 3: DeterminantsThe determinant compresses a square matrix into one number, written |A| or det A. For order two it is the main-diagonal product minus the anti-diagonal product: [3 2; 1 4] gives 3×4 − 2×1 = 10. For order three, expand along the first row: each entry times its cofactor — the second-order determinant left after deleting its row and column — with signs alternating plus, minus, plus. For example, the three first-row entries a, b, c are each multiplied by their cofactors and summed with signs, yielding a single number. A zero determinant makes the matrix singular — a term that runs through this whole lesson.
4. Step 4: Inverse matricesIf a square matrix A satisfies AB = BA = I, then B is the inverse A⁻¹; the inverse exists if and only if A is non-singular, and it is unique. Order two has a shortcut: swap the main diagonal, negate the anti-diagonal, divide by the determinant. For example, A = [2 1; 1 −3] has determinant −7, so A⁻¹ = (1/−7)[−3 −1; −1 2]. Order three goes via the adjoint: compute the nine cofactors into the cofactor matrix, transpose to get the adjoint, divide by the determinant. Properties: (A⁻¹)⁻¹ = A, (AB)⁻¹ = B⁻¹A⁻¹ and (Aᵀ)⁻¹ = (A⁻¹)ᵀ — the reversal rule matches transposition.
5. Step 5: Three methods for linear systemsWrite a system as AX = B with A the coefficient matrix. Inverse method: when A is invertible, X = A⁻¹B — multiplied from the left. Cramer's rule: x equals the determinant of A with its first column replaced by the constants, divided by D; y and z follow likewise, where D = det A. Gaussian elimination: apply row operations to the augmented matrix [A|B] until it is in reduced echelon form and read the solution. For example, a two-variable system starts by computing D to confirm solvability, then finds x and y one by one; three methods agreeing is mutual verification — in the exam use the one that flows and check with another.
6. Step 6: Singular matrices and classifying solutionsWhen D = 0, Cramer's rule fails: the system has either no solution or infinitely many — parallel lines share no point, coincident lines share all of them, and the case needs separate analysis. A homogeneous system AX = 0 always has the all-zero solution, the trivial solution; the curriculum's theorem states it has a non-trivial solution if and only if the coefficient matrix is singular. For example, x + 2y = 0 and 2x + 4y = 0 describe the same line through the origin, the coefficient determinant is zero, and every solution with x = −2y is non-trivial, such as x = 2, y = −1.
7. Step 7: Applications and presentationApplications translate a situation into AX = B: define the unknowns, write one equation per sentence, form the coefficient and constant matrices, verify the determinant is non-zero, then solve by the inverse. For example, three pens plus two notebooks cost 31 dollars, one pen plus three notebooks cost 17 dollars: the coefficient matrix [3 2; 1 3] has determinant 7, and multiplying its inverse by the constant column yields the unit prices. In your answer, state the order and the meaning of each row and column, keep the unknown and constant matrices distinct, and conclude in words with units — presentation details carry marks.

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Matrix multiplication pairs rows with columns and never commutes; AB is defined when A's columns equal B's rows.
• The determinant decides invertibility: det A = 0 means singular and no inverse; the order-two inverse swaps the main diagonal, negates the anti-diagonal, divides by det.
• Know the inverse properties: (A⁻¹)⁻¹ = A, (AB)⁻¹ = B⁻¹A⁻¹, (Aᵀ)⁻¹ = (A⁻¹)ᵀ; check the determinant before inverting.
• Three methods for systems: the inverse method X = A⁻¹B (left-multiplied), Cramer's ratios of determinants, Gaussian elimination to reduced echelon form.
• A homogeneous system always has the trivial solution and has non-trivial ones if and only if the coefficient matrix is singular; for D = 0 the non-homogeneous system has no or infinitely many solutions.