What are eigenvalues and eigenvectors?

Eigenvalues (lambda) and eigenvectors (v) satisfy the equation Av = lambda * v. They represent directions in space where a linear transformation A only scales the vector, rather than rotating it.

What does it mean if columns are linearly independent?

A set of columns is linearly independent if no column can be written as a linear combination of the others. In this case, the null space of the matrix contains only the zero vector.

Can a matrix have multiple RREFs?

No, a matrix has one and only one unique Reduced Row Echelon Form (RREF). Even if you perform row operations in a different order, the final result is identical.

What is a leading entry?

A leading entry is the first non-zero number from the left in a row of a matrix. In RREF, all leading entries must be equal to 1.

Step-by-Step Matrix Multiplication: Rules, Dimensions, and Examples

The Compatibility Rule

Unlike scalar multiplication, you cannot just multiply any two matrices together. They must satisfy a strict dimension compatibility rule: the number of columns in the first matrix must equal the number of rows in the second matrix.

If matrix A has size m × n and matrix B has size p × q, the product AB is defined if and only if:

n = p

The resulting product matrix C will have the size m × q (the outer dimensions of A and B).

The Row-by-Column Dot Product

To calculate the element in Row i, Column j of the product matrix, you take the dot product of Row i of the first matrix and Column j of the second matrix. You multiply corresponding elements and add the results together:

C[i][j] = A[i][0] × B[0][j] + A[i][1] × B[1][j] + ... + A[i][n-1] × B[n-1][j]

Worked Example

Let's multiply a 2x3 matrix A and a 3x2 matrix B:

A =

1	2	3
4	5	6

B =

7	8
9	1
0	2

The inner dimensions match (3 columns in A, 3 rows in B), so multiplication is valid. The product AB will be a 2x2 matrix C:

C[0][0] (Row 1 of A • Column 1 of B):
(1 × 7) + (2 × 9) + (3 × 0) = 7 + 18 + 0 = 25.
C[0][1] (Row 1 of A • Column 2 of B):
(1 × 8) + (2 × 1) + (3 × 2) = 8 + 2 + 6 = 16.
C[1][0] (Row 2 of A • Column 1 of B):
(4 × 7) + (5 × 9) + (6 × 0) = 28 + 45 + 0 = 73.
C[1][1] (Row 2 of A • Column 2 of B):
(4 × 8) + (5 × 1) + (6 × 2) = 32 + 5 + 12 = 49.

The product matrix C = AB is:

25	16
73	49

Key Algebraic Properties

Non-Commutative: In general, matrix multiplication is not commutative: AB ≠ BA. Swapping the order changes the row-column pairings, and often even makes the dimensions incompatible.
Identity Element: Multiplying any matrix by the Identity matrix preserves it: AI = IA = A.
Associative: (AB)C = A(BC).

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