Matrix
In short: A matrix is a rectangular grid of numbers arranged in rows and columns, used to store data and to carry out addition, scaling and multiplication in an organized way.
PrecalculusAlgebraN-VM.C.6
What is Matrix?
A matrix is a table of numbers treated as a single object. A spreadsheet of three stores and four products is one, and so is the set of coefficients in a system of equations.
Its size is written rows × columns, so a matrix with 2 rows and 3 columns is \(2 \times 3\). The entry in row \(i\) and column \(j\) is called \(a_{ij}\). Matrices of the same size add entry by entry; scalar multiplication scales every entry; and a product \(AB\) exists only when the columns of \(A\) equal the rows of \(B\).
How to spot it
- Numbers are arranged in a rectangular array inside brackets, with rows and columns.
- A table of data, like sales by store and by month, must be added or scaled as one unit.
- A system of linear equations is rewritten using a grid of coefficients.
- A transformation of points in the plane, such as a rotation, is represented by multiplying by a grid.
The method
- State the size by counting rows first and columns second.
- To add or subtract, confirm equal sizes, then combine entries in matching positions.
- To multiply, check that the inside dimensions match; each product entry is a row of the first times a column of the second.
- Verify the size of the answer: an \(m \times n\) times an \(n \times p\) gives \(m \times p\).
Matrix: solved example
Problem
Let \(A = \begin{bmatrix} 2 & -1 \\ 0 & 5 \end{bmatrix}\) and \(B = \begin{bmatrix} 3 & 4 \\ -2 & 1 \end{bmatrix}\). Find \(A + B\) and \(AB\).
Solution
Add matching entries: \(A + B = \begin{bmatrix} 5 & 3 \\ -2 & 6 \end{bmatrix}\).
For the product, row 1 of \(A\) times column 1 of \(B\) gives \(2(3) + (-1)(-2) = 8\); row 1 times column 2 gives \(2(4) + (-1)(1) = 7\); row 2 gives \(0(3) + 5(-2) = -10\) and \(0(4) + 5(1) = 5\). So \(AB = \begin{bmatrix} 8 & 7 \\ -10 & 5 \end{bmatrix}\).
Try it
Question 1
What is the size of a matrix with 3 rows and 4 columns, and how many entries does it have?
Show the answer
It is \(3 \times 4\) and has \(3 \times 4 = 12\) entries.
Question 2
Compute \(4\begin{bmatrix} 1 & -2 \\ 0 & 3 \end{bmatrix}\).
Show the answer
\(\begin{bmatrix} 4 & -8 \\ 0 & 12 \end{bmatrix}\).
Matrix vs. Determinant
The grid itself is a collection of many numbers, and it has a size such as 2 × 3. The determinant is one single number computed from a square grid. A grid can be any shape; a determinant exists only when the number of rows equals the number of columns.
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