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Algebra 2 Matrices (introductory)

Matrix operations (add, subtract, multiply)

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Theory

A matrix is a rectangular array of numbers. Operations:

  • Add / subtract entry by entry (same size only).
  • Scalar multiply: multiply every entry.
  • Multiply two matrices with row-by-column dot products; the inner dimensions must match.
Matrix multiplication is not commutative: \(AB\neq BA\) in general.
Adding matrices Matrices of the same size add entry by entry. 1 2 3 4 + 5 6 7 8 = 6 8 10 12 add entry by entry (same size)
Matrices of the same size add entry by entry.
Matrix operations Matrix operations Matrix operations add/subtract: entry by entry (same size) scalar: multiply every entry multiply: rows × columns (dot products) inner dimensions must match
The matrix operations.

Dimensions for a product:

\[(m\times n)(n\times p)=m\times p\]
the product of an m by n and n by p matrix is m by p
Inner dimensions must match; the outer ones give the result's size.

How to operate on matrices

  1. For \(\pm\), match entries (same size).
  2. For a scalar, multiply every entry.
  3. For a product, check inner dimensions match.
  4. Compute each entry as a row-column dot product.
Example 1 — Add
Add \(\begin{bmatrix}1&2\\3&4\end{bmatrix}+\begin{bmatrix}5&6\\7&8\end{bmatrix}\).
Solution

Add matching entries.

\(=\)\(\begin{bmatrix}6&8\\10&12\end{bmatrix}\)
the sum adds entry by entry
Example 2 — Scalar multiply
Find \(2\begin{bmatrix}1&2\\3&4\end{bmatrix}\).
Solution

Multiply every entry by \(2\).

\(=\)\(\begin{bmatrix}2&4\\6&8\end{bmatrix}\)
multiply every entry by 2
Example 3 — Multiply matrices
Find \(\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}\).
Solution

Multiplying by the identity leaves it unchanged.

\(=\)\(\begin{bmatrix}1&2\\3&4\end{bmatrix}\)
multiplying by the identity leaves the matrix unchanged
Example 4 — Dimension rule
Can you multiply a \(2\times3\) by a \(3\times2\) matrix?
Solution

Inner dimensions match (\(3=3\)), giving a \(2\times2\) product.

\((2\times3)(3\times2)\)\(=\)\(2\times2\)
yes, the product is 2 by 2

Common pitfalls

Add only same-size matrices.
Matrix multiplication needs matching inner dimensions.
\(AB\neq BA\) in general.

Frequently asked questions

How do you add matrices?

Add corresponding entries; the matrices must be the same size.

How do you scalar-multiply a matrix?

Multiply every entry by the scalar.

When can two matrices be multiplied?

When the inner dimensions match: \((m\times n)(n\times p)\).

Is matrix multiplication commutative?

No — \(AB\) usually does not equal \(BA\).

More in Matrices (introductory)
Solving systems using matrices
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