01 Key Concepts
Matrix Multiplication Rule
To multiply A (m x n) by B (n x p), the number of columns in A must equal the number of rows in B; the result is an m x p matrix.
How to Multiply Matrices
Each entry in the result is the sum of products of a row from A and a column from B (the 'dot product' of that row and column).
Matrix Multiplication is NOT Commutative
In general, A*B does not equal B*A — order matters, unlike with ordinary numbers.
Inverse of a Matrix (A^-1)
A matrix that 'undoes' A, so that A * A^-1 = I (the identity matrix). Only square matrices with nonzero determinant have an inverse.
Inverse of a 2x2 Matrix
For A=[[a,b],[c,d]], A^-1 = (1/det(A)) * [[d,-b],[-c,a]], where det(A) = ad-bc.
02 Key Formulas
- (A*B)_ij = sum over k of A_ik * B_kj
- For A=[[a,b],[c,d]]: A^-1 = (1/(ad-bc)) * [[d,-b],[-c,a]]
03 Solved Examples
Example 1 Multiply [[1,2],[3,4]] by [[5,6],[7,8]].
- Top-left: (1*5)+(2*7) = 5+14 = 19.
- Top-right: (1*6)+(2*8) = 6+16 = 22.
- Bottom-left: (3*5)+(4*7) = 15+28 = 43.
- Bottom-right: (3*6)+(4*8) = 18+32 = 50.
Answer: [[19,22],[43,50]]
Example 2 Find the inverse of [[2,1],[1,1]].
- Find det = (2*1)-(1*1) = 2-1 = 1.
- Apply the formula: A^-1 = (1/1)*[[1,-1],[-1,2]].
Answer: [[1,-1],[-1,2]]
Example 3 Can the matrices [[1,2,3]] (1x3) and [[4,5]] (1x2) be multiplied as A*B?
- A has 3 columns; B has 1 row.
- For A*B to be valid, A's columns must equal B's rows: 3 does not equal 1.
Answer: No, this multiplication is not defined
04 Practice Questions
1Multiply [[1,0],[0,1]] by [[3,4],[5,6]].
[[3,4],[5,6]] (multiplying by identity leaves it unchanged)
2Find the inverse of [[1,2],[3,4]].
det=-2, A^-1=[[-2,1],[1.5,-0.5]]
3Is matrix multiplication commutative in general?
No
4Can you multiply a 2x3 matrix by a 3x2 matrix?
Yes, the result is a 2x2 matrix
5Find the inverse of [[2,0],[0,2]].
[[0.5,0],[0,0.5]]
📄 Matrix Operations — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.