Operations on Matrices

3.4 Operations on Matrices

Matrices are not just static arrays of numbers; they can be combined and transformed. This section introduces the fundamental operations on matrices — addition, subtraction, scalar multiplication, and multiplication — each with rules arising from the structure of matrices themselves.

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3.4.1 Addition of Matrices

Addition is defined only when the two matrices are of the same order.

Definition (Addition of Matrices)

If $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{m \times n}$ are of the same order $m \times n$, then their sum is the matrix $C = [c_{ij}]_{m \times n}$ with

That is, add the entries in the same row and column.

Example

For $A = \begin{bmatrix} 2 & 5 \\ 1 & -3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -1 \\ 0 & 7 \end{bmatrix}$ (both $2 \times 2$),

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3.4.2 Multiplication of a Matrix by a Scalar

A scalar is a real number. Multiplying a matrix by a scalar means multiplying every entry by that scalar.

Definition (Scalar Multiplication)

If $A = [a_{ij}]_{m \times n}$ and $k$ is a scalar, then $kA = [b_{ij}]_{m \times n}$ where

Example

For $A = \begin{bmatrix} 3 & -2 \\ 0 & 5 \end{bmatrix}$ and $k = -2$,

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3.4.3 Properties of Matrix Addition and Scalar Multiplication

The following hold for any matrices $A$, $B$, $C$ of the same order $m \times n$ and any scalars $k$, $l$. They let us manipulate matrix equations like ordinary equations.

Property (I): Commutativity of Addition   $A + B = B + A$

Proof. $(A + B)_{ij} = a_{ij} + b_{ij} = b_{ij} + a_{ij} = (B + A)_{ij}$ for every entry. $\square$

Property (II): Associativity of Addition   $(A + B) + C = A + (B + C)$

Proof. $((A + B) + C)_{ij} = (a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij}) = (A + (B + C))_{ij}$, since real-number addition is associative. $\square$

Property (III): Additive Identity (Zero Matrix)   There is a zero matrix $O$ of order $m \times n$ with $A + O = A = O + A$.

Proof. $(A + O)_{ij} = a_{ij} + 0 = a_{ij}$; the other equality follows by commutativity. $\square$

Property (IV): Additive Inverse   For every $A = [a_{ij}]$ there is $-A = [-a_{ij}]$ with $A + (-A) = O = (-A) + A$.

Proof. $(A + (-A))_{ij} = a_{ij} + (-a_{ij}) = 0$. $\square$

Property (V): Distributivity over Matrix Addition   $k(A + B) = kA + kB$

Proof. $(k(A + B))_{ij} = k(a_{ij} + b_{ij}) = k a_{ij} + k b_{ij} = (kA + kB)_{ij}$. $\square$

Property (VI): Distributivity over Scalar Addition   $(k + l)A = kA + lA$

Proof. $((k + l)A)_{ij} = (k + l)a_{ij} = k a_{ij} + l a_{ij} = (kA + lA)_{ij}$. $\square$

Property (VII): Associativity of Scalar Multiplication   $k(lA) = (kl)A$

Proof. $(k(lA))_{ij} = k(l a_{ij}) = (kl)a_{ij} = ((kl)A)_{ij}$. $\square$

Property (VIII): Multiplication by 1   $1 \cdot A = A$

Proof. $(1 \cdot A)_{ij} = 1 \cdot a_{ij} = a_{ij}$. $\square$

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3.4.4 Difference of Matrices

Definition (Difference of Matrices)

If $A$ and $B$ are of the same order $m \times n$, their difference is

Example

For $A = \begin{bmatrix} 5 & 3 \\ -1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 7 \\ 4 & -3 \end{bmatrix}$,

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3.4.5 Multiplication of Matrices

Matrix multiplication is not defined entry-wise; it corresponds to composition of linear transformations.

Definition (Multiplication of Matrices)

Let $A = [a_{ij}]$ be $m \times n$ and $B = [b_{jk}]$ be $n \times p$. The product $AB$ is the $m \times p$ matrix $C = [c_{ik}]$ whose entry $c_{ik}$ is the dot product of the $i$-th row of $A$ with the $k$-th column of $B$:

Example

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ ($2 \times 2$) and $B = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}$ ($2 \times 3$). Since $A$ has 2 columns and $B$ has 2 rows, $AB$ is defined and is $2 \times 3$. Taking each row of $A$ against each column of $B$:

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3.4.6 Properties of Matrix Multiplication

Property (IX): Associativity   If $A$ is $m \times n$, $B$ is $n \times p$, $C$ is $p \times q$, then $(AB)C = A(BC)$.

Proof. The $(i, l)$ entry of $(AB)C$ is $\sum_{k=1}^{p}\left(\sum_{j=1}^{n} a_{ij} b_{jk}\right) c_{kl} = \sum_{k}\sum_{j} a_{ij} b_{jk} c_{kl}$, and that of $A(BC)$ is $\sum_{j=1}^{n} a_{ij}\left(\sum_{k=1}^{p} b_{jk} c_{kl}\right) = \sum_{j}\sum_{k} a_{ij} b_{jk} c_{kl}$. The double sums agree. $\square$

Property (X): Distributivity over Addition   $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$ (for conformable orders).

Proof (first part). The $(i, k)$ entry of $A(B+C)$ is $\sum_{j} a_{ij}(b_{jk} + c_{jk}) = \sum_{j} a_{ij} b_{jk} + \sum_{j} a_{ij} c_{jk} = (AB)_{ik} + (AC)_{ik}$. The second part is similar. $\square$ …