Question 1

When can you add or subtract two matrices?

Accepted Answer

Two matrices can only be added or subtracted when they have the same dimensions (same number of rows and same number of columns). You simply add (or subtract) the corresponding elements: (A±B)ᵢⱼ = aᵢⱼ ± bᵢⱼ. A 2×3 matrix cannot be added to a 3×2 matrix; they must both be 2×3.

Question 2

What are the rules for matrix multiplication?

Accepted Answer

Matrix multiplication A×B requires A to have the same number of columns as B has rows. If A is m×n and B is n×p, the product AB is m×p. Each entry (AB)ᵢⱼ is the dot product of row i of A with column j of B. Matrix multiplication is NOT commutative — AB ≠ BA in general.

Question 3

What is the transpose of a matrix?

Accepted Answer

The transpose Aᵀ is formed by flipping rows and columns: the first row becomes the first column, and so on. If A is m×n, then Aᵀ is n×m. A symmetric matrix satisfies A = Aᵀ. The transpose appears in many formulas: (AB)ᵀ = BᵀAᵀ, and the dot product of vectors u·v = uᵀv.

Question 4

Is matrix multiplication associative and distributive?

Accepted Answer

Yes. Matrix multiplication is associative: (AB)C = A(BC). It is also distributive: A(B+C) = AB + AC. However, it is NOT commutative: AB ≠ BA in general. The identity matrix I acts like the number 1: AI = IA = A. The zero matrix acts like 0: A + 0 = A and A × 0 = 0.

Question 5

What real-world problems use matrix multiplication?

Accepted Answer

Matrix multiplication solves systems of linear equations (Ax = b), transforms coordinates in computer graphics (rotation, scaling, translation), represents neural network layers in deep learning, models Markov chains in probability theory, and compresses images using Singular Value Decomposition (SVD). Any linear transformation between vector spaces is represented by a matrix.

Question 6

What is the identity matrix and why does it matter?

Accepted Answer

The identity matrix I is a square matrix with 1s on the main diagonal and 0s elsewhere. For any matrix A: AI = IA = A. It is the matrix equivalent of multiplying by 1. The 2×2 identity is [[1,0],[0,1]]; the 3×3 identity is the 3×3 analog. When you solve Ax = b using A⁻¹, the key property used is A⁻¹A = I.

Operation	Formula	Condition	Result Size
Addition A + B	(A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ	A and B same size	Same as A and B
Subtraction A − B	(A−B)ᵢⱼ = aᵢⱼ − bᵢⱼ	A and B same size	Same as A and B
Multiplication A × B	(AB)ᵢⱼ = Σ aᵢₖ·bₖⱼ	A: m×n, B: n×p	m×p matrix
Transpose Aᵀ	(Aᵀ)ᵢⱼ = aⱼᵢ	Any matrix A	n×m if A is m×n
Scalar multiplication cA	(cA)ᵢⱼ = c·aᵢⱼ	Any scalar c	Same as A
Identity I	Diagonal 1s, rest 0s	—	AI = IA = A

Position	Dot Product	Calculation	Result
R1C1	Row 1 · Col 1	(1)(5) + (2)(7)	19
R1C2	Row 1 · Col 2	(1)(6) + (2)(8)	22
R2C1	Row 2 · Col 1	(3)(5) + (4)(7)	43
R2C2	Row 2 · Col 2	(3)(6) + (4)(8)	50

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