Matrix Operations

Grade 11 · Math · Study Guide

A comprehensive guide to matrix arithmetic, determinants, inverses, and solving linear systems using matrices.

A matrix is a rectangular array of numbers arranged in rows and columns. An m×n matrix has m rows and n columns. Matrices support addition and subtraction when they have the same dimensions, performed element by element. Scalar multiplication multiplies every element by the scalar. Matrix multiplication of A (m×n) by B (n×p) yields a C (m×p) matrix; element C_ij is the dot product of row i of A with column j of B. Matrix multiplication is NOT commutative: AB ≠ BA in general.

The determinant of a 2×2 matrix [[a,b],[c,d]] is det(A) = ad - bc. The inverse of a 2×2 matrix exists only if det(A) ≠ 0, and is A⁻¹ = (1/det(A)) * [[d,-b],[-c,a]]. For larger matrices, the inverse can be found using row reduction (Gauss-Jordan elimination).

Systems of linear equations can be written as AX = B. If A is invertible, the unique solution is X = A⁻¹B. Alternatively, Cramer's Rule uses determinants to solve for each variable: x_i = det(A_i)/det(A), where A_i replaces the ith column with the constant vector B. Matrices are foundational in computer graphics, engineering, economics, and all fields involving simultaneous equations.

Key terms

Matrix
A rectangular array of numbers organized into rows and columns, denoted m×n for m rows and n columns.
Determinant
A scalar value computed from a square matrix that indicates whether the matrix is invertible; det([[a,b],[c,d]]) = ad-bc.
Inverse Matrix
Matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I (identity matrix); exists only if det(A) ≠ 0.
Identity Matrix
A square matrix with 1s on the diagonal and 0s elsewhere; acts as the multiplicative identity: AI = IA = A.
Row Reduction
Systematic elimination process (Gaussian elimination) used to solve systems of equations or find matrix inverses.
Cramer's Rule
Method using determinants to solve Ax = b; x_i = det(A_i)/det(A), replacing the ith column with vector b.

Flashcards

Matrix dimensions
m×n means m rows and n columns; element a_ij is in row i, column j
Matrix multiplication (AB)_ij
Sum of (row i of A) dot (column j of B); requires inner dimensions to match
2×2 determinant
det([[a,b],[c,d]]) = ad - bc
2×2 inverse
A⁻¹ = (1/(ad-bc)) * [[d,-b],[-c,a]]
Singular matrix
A square matrix with det = 0; non-invertible, corresponding system has no unique solution
Cramer's Rule
x_i = det(A_i)/det(A); replace ith column of A with constant vector b
Augmented matrix
[A|b] used in Gaussian elimination to solve Ax = b
Row echelon form
Matrix form with zeros below pivots, used for solving linear systems by back-substitution

Questions

  1. If A is a 2×3 matrix and B is a 3×4 matrix, what is the dimension of AB?

    1. 3×3
    2. 2×4
    3. 3×4
    4. 2×3
    Show answer

    Answer: 2×4

    A (2×3) times B (3×4): outer dimensions give 2×4.

  2. The determinant of [[5, 2], [3, 4]] is ___.

    Show answer

    Answer: 14

    det = (5)(4) - (2)(3) = 20 - 6 = 14.

  3. Matrix multiplication is commutative; that is, AB = BA for all matrices A and B.

    Show answer

    Answer: False

    Matrix multiplication is generally NOT commutative; AB ≠ BA in general.

  4. What is the inverse of the matrix [[2, 1], [5, 3]]?

    1. [[3, -1], [-5, 2]]
    2. [[-3, 1], [5, -2]]
    3. [[3, 1], [5, 2]]
    4. [[1/2, 0], [0, 1/3]]
    Show answer

    Answer: [[3, -1], [-5, 2]]

    det = (2)(3)-(1)(5) = 1. A⁻¹ = (1/1)[[3,-1],[-5,2]] = [[3,-1],[-5,2]].

  5. If det(A) = 0, then the matrix A has no inverse.

    Show answer

    Answer: True

    A matrix is invertible if and only if its determinant is nonzero. det=0 means it is singular (no inverse).

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