Matrices
Get started with matrices for Victorian Year 11 Mathematical Methods (VCAA) — rectangular arrays of numbers used to store and combine information and to describe transformations.
You will learn matrix order and notation, add and subtract matrices, multiply by a scalar, and solve for unknown entries using equality — the foundation for using matrices with transformations of the plane.
Theory
In Year 11 Mathematical Methods (Unit 1), a matrix is a rectangular array of numbers set out in rows and columns. This page shows how to state a matrix's order (dimension), read an element \(a_{ij}\), multiply a matrix by a scalar, add and subtract matrices of the same order, and use equal matrices to solve for unknowns.
A matrix is a rectangular array of numbers arranged in horizontal rows and vertical columns, written inside square brackets, e.g. \(A=\begin{bmatrix} 4 & -1 \\ 0 & 3 \\ 2 & 5 \end{bmatrix}\).
The order (or dimension) of a matrix is rows \(\times\) columns, in that order. The matrix \(A\) above has \(3\) rows and \(2\) columns, so its order is \(3\times 2\). A matrix of order \(m\times n\) has \(m\times n\) elements in total.
Each element is named \(a_{ij}\), the entry in row \(i\), column \(j\) (row first, column second).
Scalar multiplication: to multiply a matrix by a number (a scalar), multiply every element by that number.
Addition and subtraction: add or subtract two matrices of the same order by combining corresponding elements. If the orders differ, the operation is undefined.
Equal matrices: two matrices are equal only when they have the same order and every pair of corresponding elements is equal — this lets you form equations and solve for unknowns.
For matrices of the same order, with scalar \(k\):
Working with matrices
- Order (dimension): count the rows, then the columns, and write it as rows \(\times\) columns.
- Element \(a_{ij}\): go to row \(i\), then column \(j\), and read the entry.
- Scalar multiple \(kA\): multiply every element of \(A\) by \(k\).
- Add or subtract: check the orders match, then combine corresponding elements one position at a time.
- Equal matrices: set corresponding elements equal to form equations, then solve for the unknowns.
(a) Order of \(A\) — count the rows, then the columns:
| \(\text{rows}\) | \(=\) | \(3\) |
| \(\text{columns}\) | \(=\) | \(2\) |
Order is written rows \(\times\) columns, so \(A\) is \(3\times 2\).
(b) Element \(b_{23}\) — go to row \(2\), column \(3\) of \(B\):
| \(b_{23}\) | \(=\) | \(-3\) |
\(A\) has order \(3\times 2\), and \(b_{23}=-3\).
Multiply every element of \(A\) by the scalar \(3\):
| \(3A\) | \(=\) | \(3\begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix}\) |
| \(=\) | \(\begin{bmatrix} 3\times 2 & 3\times(-1) \\ 3\times 4 & 3\times 0 \end{bmatrix}\) | |
| \(=\) | \(\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}\) |
\(3A=\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}\).
Scalar multiply each matrix first:
| \(3A\) | \(=\) | \(\begin{bmatrix} 6 & 3 \\ 0 & -9 \end{bmatrix}\) |
| \(2B\) | \(=\) | \(\begin{bmatrix} 2 & -4 \\ 8 & 10 \end{bmatrix}\) |
Subtract corresponding elements \((3A-2B)\):
| \(3A-2B\) | \(=\) | \(\begin{bmatrix} 6-2 & 3-(-4) \\ 0-8 & -9-10 \end{bmatrix}\) |
| \(=\) | \(\begin{bmatrix} 4 & 7 \\ -8 & -19 \end{bmatrix}\) |
\(3A-2B=\begin{bmatrix} 4 & 7 \\ -8 & -19 \end{bmatrix}\).
Equal matrices have equal corresponding elements, so equate the \((1,1)\) and \((2,2)\) entries.
Form two equations:
| \(x+y\) | \(=\) | \(10 \quad (1)\) |
| \(x-y\) | \(=\) | \(2 \quad (2)\) |
(i) Add \((1)\) and \((2)\) to eliminate \(y\):
| \((x+y)+(x-y)\) | \(=\) | \(10+2\) |
| \(2x\) | \(=\) | \(12\) |
| \(x\) | \(=\) | \(6\) |
(ii) Substitute \(x=6\) into \((1)\):
| \(6+y\) | \(=\) | \(10\) |
| \(y\) | \(=\) | \(4\) |
\(x=6,\ y=4\).
Common pitfalls
Frequently asked questions
What is the order of a matrix?
The order (dimension) is the number of rows by the number of columns, written rows \(\times\) columns. A matrix with \(3\) rows and \(2\) columns has order \(3\times 2\).
What does the notation a_{ij} mean?
\(a_{ij}\) is the element in row \(i\) and column \(j\) of matrix \(A\) — the row index comes first, the column index second.
When can you add or subtract two matrices?
Only when they have the same order (same number of rows and same number of columns). You then add or subtract corresponding elements.
How do you multiply a matrix by a scalar?
Multiply every element of the matrix by the scalar. For example, \(3\begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix}=\begin{bmatrix} 6 & -3 \\ 12 & 0 \end{bmatrix}\).
When are two matrices equal?
Two matrices are equal when they have the same order and every pair of corresponding elements is equal. Equating corresponding elements gives equations you can solve for unknowns.