Key takeaway: A matrix is a grid of numbers organized into rows and columns and sized as m × n, used to store tables of data and to transform vectors.
Grade 12 – PrecalculusN-VM.C.6Expressions and Equations
Definition of Matrix
A matrix is a table of numbers with a fixed shape. Each number is an entry, each horizontal line is a row, and each vertical line is a column. One with 3 rows and 2 columns is called a 3 by 2 table, always with the row count first.
Grids of this kind store data such as prices by store, and they also act on points and vectors: multiplying by one can rotate, stretch or reflect a figure. Adding needs two grids of identical size. Multiplying is different, requiring the number of columns in the first factor to equal the number of rows in the second.
Formula and notation
Entries are named by row, then column, so \(a_{23}\) lives in row 2, column 3. A 2 by 2 example:
\[A=\begin{bmatrix}2 & -1\\ 0 & 3\end{bmatrix}.\]
The product \(AB\) has entries found by pairing a row of \(A\) with a column of \(B\), multiplying and adding.
Key facts
- Two grids can be added only when their sizes match exactly.
- Reversing the order of a product usually changes the result.
- Multiplying an m × n grid by an n × p grid yields size m × p.
- The identity has 1s on the diagonal and 0s elsewhere, and leaves others unchanged.
Matrix example
Let \(A=\begin{bmatrix}2 & -1\\ 0 & 3\end{bmatrix}\) and \(B=\begin{bmatrix}4 & 5\\ 1 & -2\end{bmatrix}\). Find \(AB\).
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Top-left: row 1 of A with column 1 of B gives \(2(4)+(-1)(1)=7\).
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Top-right: row 1 with column 2 gives \(2(5)+(-1)(-2)=12\).
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Bottom-left: \(0(4)+3(1)=3\). Bottom-right: \(0(5)+3(-2)=-6\).
\(AB=\)\(\begin{bmatrix}7 & 12\\ 3 & -6\end{bmatrix}\).
Common mistake
Watch out
Jiro assumes \(BA\) must equal \(AB\), so he reuses the answer above.
Why it is wrong
Matrix multiplication depends on order. Computing \(BA\) row by row gives \(\begin{bmatrix}8 & 11\\ 2 & -7\end{bmatrix}\), which differs from \(AB\) in three of four entries.
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Expressions and Equations