11.5 Matrices and Matrix Operations

Matrix Operations

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices give us a compact way to organize data and, more importantly for this unit, to represent and solve systems of linear equations. This section covers the core operations you can perform on matrices and how those operations connect back to solving systems.

Matrix Addition and Subtraction

Two matrices can only be added or subtracted if they share the same dimensions (same number of rows and same number of columns). When that condition is met, you simply add or subtract the corresponding entries, position by position. The result is a new matrix with those same dimensions.

$\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} + \begin{bmatrix}5 & 6\\7 & 8\end{bmatrix} = \begin{bmatrix}1+5 & 2+6\\3+7 & 4+8\end{bmatrix} = \begin{bmatrix}6 & 8\\10 & 12\end{bmatrix}$

Subtraction works the same way: subtract each entry in the second matrix from the corresponding entry in the first.

Scalar multiplication means multiplying every entry in a matrix by a single constant (the scalar). The resulting matrix keeps the same dimensions.

$3 \times \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} = \begin{bmatrix}3(1) & 3(2)\\3(3) & 3(4)\end{bmatrix} = \begin{bmatrix}3 & 6\\9 & 12\end{bmatrix}$

Key properties of addition and scalar multiplication:

• Commutative property of addition: $A + B = B + A$
• Associative property of addition: $(A + B) + C = A + (B + C)$
• Distributive property of scalar multiplication over addition: $k(A + B) = kA + kB$

Matrix Multiplication

Matrix multiplication is more involved than addition. Two conditions and a process to remember:

Dimension requirement: To multiply matrix $A$ (dimensions $m \times n$) by matrix $B$ (dimensions $n \times p$), the number of columns in $A$ must equal the number of rows in $B$. The resulting matrix will have dimensions $m \times p$.

How to find each entry of the product:

1. Pick a row from the first matrix and a column from the second matrix.
2. Multiply each entry in that row by the corresponding entry in that column.
3. Add up all those products. That sum is one entry in the result matrix.
4. Repeat for every row-column combination.

$\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} \times \begin{bmatrix}5 & 6\\7 & 8\end{bmatrix} = \begin{bmatrix}(1)(5)+(2)(7) & (1)(6)+(2)(8)\$3)(5)+(4)(7) & (3)(6)+(4)(8)\end{bmatrix} = \begin{bmatrix}19 & 22\\43 & 50\end{bmatrix}$

For example, the entry in row 1, column 1 of the result is $(1)(5) + (2)(7) = 5 + 14 = 19$.

Properties of matrix multiplication:

• Not commutative: In general, $AB \neq BA$. This is a big difference from regular number multiplication, and a common exam mistake.
• Associative: $(AB)C = A(BC)$
• Distributive over addition: $A(B + C) = AB + AC$

The identity matrix is a square matrix with 1s along the main diagonal and 0s everywhere else. For a $2 \times 2$ matrix, it looks like this:

$I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$

Multiplying any matrix by the appropriately sized identity matrix returns the original matrix: $AI = IA = A$. Think of it as the matrix equivalent of multiplying a number by 1.

Matrices in Systems of Equations

Any system of linear equations can be written in matrix form $AX = B$, where:

• $A$ = the coefficient matrix (the coefficients of the variables)
• $X$ = the variable matrix (a column matrix of the unknowns)
• $B$ = the constant matrix (the numbers on the right side of each equation)

For example, the system $2x + 3y = 7$ and $x - y = 1$ becomes:

$\begin{bmatrix}2 & 3\\1 & -1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}7\\1\end{bmatrix}$

Solving using the inverse matrix:

1. Write the system in matrix form: $AX = B$.
2. Find the inverse of the coefficient matrix, $A^{-1}$ (this exists only if $A$ is square and its determinant is not zero).
3. Multiply both sides on the left by $A^{-1}$: $A^{-1}AX = A^{-1}B$.
4. Since $A^{-1}A = I$ and $IX = X$, you get $X = A^{-1}B$.

A square matrix $A$ has an inverse $A^{-1}$ when $AA^{-1} = A^{-1}A = I$. Common methods for finding the inverse include row reduction (augmenting $A$ with $I$ and reducing $A$ to $I$) and, for $2 \times 2$ matrices, using the determinant formula. If the determinant of $A$ equals zero, the matrix has no inverse and the system either has no solution or infinitely many solutions.

Matrix Properties

• Dimension: Written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns. A $3 \times 2$ matrix has 3 rows and 2 columns.
• Transpose: The transpose of matrix $A$, written $A^T$, flips the matrix over its main diagonal, turning rows into columns and columns into rows. If $A$ is $m \times n$, then $A^T$ is $n \times m$.