Lesson 2: Properties of Matrix Arithmetic and Transpose

Estimated time: 30-40 minutes

Learning Objectives

State and apply the algebraic properties of matrix addition and multiplication
Understand why matrix multiplication is NOT commutative
Compute the transpose of a matrix
Apply properties of the transpose, including (AB)^T = B^T A^T
Identify symmetric matrices

Properties of Matrix Addition

For matrices A, B, C of the same size and scalar c:

A + B = B + A (commutative)
(A + B) + C = A + (B + C) (associative)
A + O = A where O is the zero matrix
A + (-A) = O
c(A + B) = cA + cB (scalar distributes over addition)
(c + d)A = cA + dA

Properties of Matrix Multiplication

For matrices of compatible sizes:

A(BC) = (AB)C (associative)
A(B + C) = AB + AC (left distributive)
(A + B)C = AC + BC (right distributive)
AI = A and IA = A (identity)
c(AB) = (cA)B = A(cB)

        Warnings: What Does NOT Work
        AB does not equal BA in general! Matrix multiplication is not commutative.
AB = 0 does NOT imply A = 0 or B = 0. You cannot "cancel" like with real numbers.
AB = AC does NOT imply B = C. You cannot "cancel" A from both sides.

      

Example: AB is not equal to BA

A = [ 1 2 ] B = [ 0 1 ]
[ 0 0 ] [ 0 0 ]

AB = [ 0 1 ; 0 0 ] but BA = [ 0 0 ; 0 0 ]. These are clearly different.

Example: AB = 0 but A and B are not zero

A = [ 1 0 ] B = [ 0 0 ]
[ 0 0 ] [ 3 4 ]

AB = [ 0 0 ; 0 0 ] = O. Neither A nor B is the zero matrix, yet their product is zero!

The Transpose

Transpose: The transpose of an m x n matrix A, written A^T, is the n x m matrix whose rows are the columns of A. That is, (A^T)_{ij} = a_{ji}.

Example

A = [ 1 2 3 ]     A^T = [ 1 4 ]
    [ 4 5 6 ]           [ 2 5 ]
                          [ 3 6 ]

A is 2x3; A^T is 3x2. Row 1 of A becomes column 1 of A^T.

Properties of Transpose:

(A^T)^T = A
(A + B)^T = A^T + B^T
(cA)^T = cA^T
(AB)^T = B^T A^T (order reverses!)

The Shoe-Sock Rule

When transposing a product, the order reverses: (AB)^T = B^T A^T. Think of it like putting on shoes and socks: you put socks on first, but take shoes off first (reverse order).

Symmetric Matrices

Symmetric Matrix: A square matrix A such that A^T = A. That is, a_{ij} = a_{ji} for all i, j. The matrix is symmetric about its main diagonal.

Example

A = [ 1 3 5 ]
[ 3 2 4 ]
[ 5 4 7 ]

A^T = A, so A is symmetric. Notice the entries mirror across the main diagonal.

Key fact: For any matrix A (not necessarily square), A^T A is always symmetric. Proof: (A^T A)^T = A^T (A^T)^T = A^T A.

Check Your Understanding

1. If A is 2x3 and B is 3x2, what are the sizes of AB and BA?

Answer: AB is 2x2. BA is 3x3. Both products are defined but have different sizes -- they cannot possibly be equal.

2. Find A^T if A = [ 1 -2 ; 3 4 ; 0 5 ].

Answer: A is 3x2, so A^T is 2x3: A^T = [ 1 3 0 ; -2 4 5 ].

3. If (AB)^T = ?, express it using transposes of A and B.

Answer: (AB)^T = B^T A^T. The order reverses.

4. True or False: If AB = AC then B = C.

Answer: False. You cannot cancel matrices from both sides in general. Cancellation only works if A is invertible (which we study in the next lesson).

Key Takeaways

Matrix multiplication is associative and distributive but NOT commutative
You cannot cancel matrices: AB = 0 does not imply A = 0 or B = 0
The transpose flips rows and columns: (A^T)_{ij} = a_{ji}
(AB)^T = B^T A^T -- the shoe-sock rule (order reverses)
A symmetric matrix satisfies A^T = A; the product A^T A is always symmetric

Next Lesson

Lesson 3: The Inverse of a Matrix.

Start Lesson 3

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