Matrices
Grade 11 · Math · Study Guide
Understand matrix addition, multiplication, determinants, and how to use matrices to solve linear systems.
A matrix is a rectangular array of numbers in rows and columns. An m x n matrix has m rows and n columns. Matrices are added element-by-element; the matrices must share the same dimensions.
Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. The product of an m x n and n x p matrix is m x p. Note that multiplication is not commutative: AB does not equal BA in general.
The determinant of a 2x2 matrix [[a,b],[c,d]] is ad - bc. A nonzero determinant means the matrix is invertible, enabling you to use the inverse or Cramer's Rule to solve linear systems.
| Operation | Requirement | Result Size |
|---|---|---|
| Addition A + B | Same dimensions | Same as A and B |
| Scalar multiplication kA | None | Same as A |
| Multiplication AB | cols(A) = rows(B) | m x p if A is m x n and B is n x p |
| Determinant | Square matrix | Single scalar |
| Matrix | Inverse |
|---|---|
| A = [[a,b],[c,d]] | A^(-1) = (1/det(A)) * [[d,-b],[-c,a]] |
| Condition | det(A) = ad - bc != 0 |
Key terms
- Dimensions
- Written rows x columns (m x n); a 3x2 matrix has 3 rows and 2 columns.
- Determinant
- A scalar from a square matrix; det([[a,b],[c,d]]) = ad - bc.
- Identity matrix
- Square matrix with 1s on the main diagonal and 0s elsewhere; AI = IA = A.
- Inverse matrix
- A^(-1) such that A * A^(-1) = I; exists only when det(A) != 0.
Practice questions
What is the size of the product of a 2x3 matrix multiplied by a 3x4 matrix?
- 3x3
- 2x4
- 3x4
- 2x3
Show answer
Product of m x n and n x p is m x p: (2x3)(3x4) = 2x4.
Find the determinant of [[5, 2], [3, 4]].
Show answer
det = (5)(4) - (2)(3) = 20 - 6 = 14.
Add: [[1,2],[3,4]] + [[5,6],[7,8]].
- [[6,8],[10,12]]
- [[5,12],[21,32]]
- [[6,8],[10,11]]
- [[4,4],[4,4]]
Show answer
Add element-by-element: [1+5, 2+6; 3+7, 4+8] = [6,8;10,12].
Is matrix multiplication commutative?
- Yes, always
- No, in general AB != BA
- Only for square matrices
- Only when det != 0
Show answer
Matrix multiplication is not commutative; the order of multiplication matters.
Find the inverse of A = [[3, 1],[2, 1]].
Show answer
det(A) = 3(1) - 1(2) = 1. A^(-1) = (1/1)*[[1,-1],[-2,3]] = [[1,-1],[-2,3]].
Fun facts
- Computer graphics rely heavily on matrix multiplication to rotate, scale, and translate 3D objects on screen in real time.
- Google's original PageRank algorithm was solved using a massive matrix called the adjacency matrix of the web graph.
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