4: Matrix Operations

Matrices aren’t just grids of numbers,they’re objects with their own arithmetic. Matrix addition and scalar multiplication work entry-by-entry, but matrix multiplication is something stranger and more powerful: it’s function composition in disguise.

Matrix Addition

(Matrix Addition)

Two matrices $A$ and $B$ can be added if and only if they have the same dimensions. If $A$ and $B$ are both $m \times n$, their sum is defined entry-by-entry:

Example:

(Properties of Matrix Addition)

For matrices $A$, $B$, $C$ of the same size:

1. Commutativity: $A + B = B + A$
2. Associativity: $(A + B) + C = A + (B + C)$
3. Identity: $A + O = A$ where $O$ is the zero matrix
4. Inverse: $A + (-A) = O$

Matrix addition inherits all the nice properties of real number addition.

Scalar Multiplication

(Scalar Multiplication)

For a scalar $c \in \mathbb{R}$ and matrix $A$:

Every entry gets multiplied by $c$.

Example:

(Properties of Scalar Multiplication)

For scalars $c, d$ and matrices $A, B$:

1. Associativity: $c(dA) = (cd)A$
2. Distributivity over matrix addition: $c(A + B) = cA + cB$
3. Distributivity over scalar addition: $(c + d)A = cA + dA$
4. Identity: $1 \cdot A = A$

Matrix Multiplication

Matrix multiplication is where things get interesting. Unlike addition, it’s not entry-by-entry,it encodes something deeper.

(When Can You Multiply?)

You can compute $AB$ if and only if:

If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$.

The inner dimensions must match; the outer dimensions give the result size.

(The Entry Formula)

The $(i, j)$ entry of $AB$ is computed as:

This is the dot product of row $i$ of $A$ with column $j$ of $B$.

Example:

Three Ways to See Matrix Multiplication

The entry formula is correct but unilluminating. Here are three better perspectives.

(View 1: Column Combinations)

Each column of $AB$ is $A$ times the corresponding column of $B$:

Interpretation: $B$ tells you how to take linear combinations of $A$‘s columns.

(View 2: Row Combinations)

Each row of $AB$ is the corresponding row of $A$ times $B$:

Interpretation: $A$ tells you how to take linear combinations of $B$‘s rows.

(View 3: Function Composition)

If $T_A(\mathbf{x}) = A\mathbf{x}$ and $T_B(\mathbf{x}) = B\mathbf{x}$, then:

Matrix multiplication is function composition.

The matrix $AB$ represents “first apply $B$, then apply $A$.” This is why multiplication isn’t commutative,the order of transformations matters.

(View 4: Outer Product Sum)

$AB$ can be written as a sum of rank-1 matrices:

Each term is an outer product,a column times a row,giving a rank-1 matrix. The sum builds up the full product.

Matrix Multiplication is NOT Commutative

(Non-Commutativity)

In general, $AB \neq BA$.

Reasons:

1. $AB$ might exist when $BA$ doesn’t (dimension mismatch)
2. Even when both exist, they may have different sizes
3. Even when both are the same size, the entries usually differ

Example:

Same matrices, different order, different result.

Intuition: “Rotate then reflect” is not the same as “reflect then rotate.”

Properties of Matrix Multiplication

(What DOES Hold)

For matrices of compatible sizes:

1. Associativity: $(AB)C = A(BC)$
2. Distributivity: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
3. Scalar compatibility: $c(AB) = (cA)B = A(cB)$
4. Identity: $I_m A = A = A I_n$ for $A$ of size $m \times n$

Associativity is remarkable,it says we can chain transformations without worrying about grouping. This is why we can write $ABC$ without parentheses.

(The Identity Matrix)

The identity matrix $I_n$ is the $n \times n$ matrix with $1$s on the diagonal and $0$s elsewhere:

Property: $I_n \mathbf{x} = \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^n$.

The identity matrix is the “do nothing” transformation.

The Transpose

(Transpose)

The transpose of $A$, denoted $A^T$, swaps rows and columns:

If $A$ is $m \times n$, then $A^T$ is $n \times m$.

Example:

(Properties of Transpose)

1. $(A^T)^T = A$
2. $(A + B)^T = A^T + B^T$
3. $(cA)^T = cA^T$
4. $(AB)^T = B^T A^T$ (note the order reversal!)

The transpose reversal $(AB)^T = B^T A^T$ mirrors how function composition reverses: to undo “first $B$, then $A$,” you undo $A$ first, then $B$.

(Symmetric Matrices)

A matrix is symmetric if $A^T = A$.

This means $A_{ij} = A_{ji}$,the matrix equals its mirror across the diagonal.

Example:

Symmetric matrices have special properties: their eigenvalues are real, and they can be orthogonally diagonalized.

Matrix Powers

(Powers of Square Matrices)

For a square matrix $A$ and positive integer $k$:

By convention, $A^0 = I$.

Interpretation: $A^k$ represents applying the transformation $A$ a total of $k$ times.

(Example: Powers Reveal Structure)

Then $A^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

This matrix is nilpotent,some power of it is zero. Geometrically, applying it twice collapses everything.

Matrix Inverses

(Inverse Matrix)

An $n \times n$ matrix $A$ is invertible (or nonsingular) if there exists a matrix $A^{-1}$ such that:

The matrix $A^{-1}$ is called the inverse of $A$.

Interpretation: If $A$ represents a transformation, then $A^{-1}$ is the transformation that undoes it. Applying $A$ then $A^{-1}$ (or vice versa) returns you to where you started.

(Uniqueness)

If $A$ is invertible, its inverse is unique.

Proof: Suppose $B$ and $C$ are both inverses of $A$. Then:

(2×2 Inverse Formula)

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ with $\det(A) = ad - bc \neq 0$:

Swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant.

Example:

(Computing Inverses via Row Reduction)

For larger matrices, use the augmented matrix method:

1. Form the augmented matrix $[A \mid I]$
2. Row reduce until the left side becomes $I$
3. The right side becomes $A^{-1}$

If $A$ cannot be reduced to $I$ (a row of zeros appears on the left), then $A$ is not invertible.

Example:

So $\begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$.

(When is a Matrix Invertible?)

For an $n \times n$ matrix $A$, the following are equivalent:

1. $A$ is invertible
2. $\det(A) \neq 0$
3. $\text{rank}(A) = n$
4. $\text{rref}(A) = I_n$
5. The columns of $A$ are linearly independent
6. The columns of $A$ span $\mathbb{R}^n$
7. $A\mathbf{x} = \mathbf{b}$ has exactly one solution for every $\mathbf{b}$
8. $A\mathbf{x} = \mathbf{0}$ has only the trivial solution
9. $\ker(A) = \{\mathbf{0}\}$

These are all ways of saying $A$ doesn’t collapse any dimension.

(Properties of Inverses)

For invertible matrices $A$ and $B$:

1. $(A^{-1})^{-1} = A$
2. $(AB)^{-1} = B^{-1}A^{-1}$ (note the order reversal!)
3. $(A^T)^{-1} = (A^{-1})^T$
4. $(cA)^{-1} = \frac{1}{c}A^{-1}$ for $c \neq 0$
5. $(A^k)^{-1} = (A^{-1})^k$

The reversal in $(AB)^{-1} = B^{-1}A^{-1}$ makes sense: to undo “first $B$, then $A$,” you must undo $A$ first, then undo $B$.

(Solving Systems with Inverses)

If $A$ is invertible, the system $A\mathbf{x} = \mathbf{b}$ has the unique solution:

Proof: Multiply both sides of $A\mathbf{x} = \mathbf{b}$ by $A^{-1}$:

Note: In practice, row reduction is more efficient than computing $A^{-1}$ explicitly.

(Singular Matrices)

A matrix that is not invertible is called singular.

Singular matrices have $\det(A) = 0$, meaning they collapse at least one dimension. There’s no way to “uncollapse”,information is lost, so no inverse exists.

Example:

The columns are linearly dependent (second is twice the first), $\det(A) = 0$, and $A$ maps all of $\mathbb{R}^2$ onto a line. No inverse exists.

Why Matrix Multiplication Works This Way

The definition of matrix multiplication seems arbitrary until you realize it’s forced by the requirement that matrices represent linear transformations.

If we want $(AB)\mathbf{x} = A(B\mathbf{x})$ to hold for all $\mathbf{x}$, there’s only one possible definition for $AB$. The entry formula, the column view, and all the properties follow inevitably.

Matrix multiplication is function composition,the rest is just computing what that means entry-by-entry.