Mathfolis

Matrices

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

A matrix is a rectangular array of numbers. Addition and scalar multiplication act entry-by-entry. Matrix multiplication is more involved: the (i,j)(i, j) entry of ABAB is the dot product of row ii of AA with column jj of BB. Matrix-vector multiplication is a special case where the second matrix is a column vector.

2×2 multiplication formula

Product
(abcd)(efgh)=(ae+bgaf+bhce+dgcf+dh)\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}
Caution: Matrix multiplication is not commutative. In general, AB≠BAAB \neq BA.
Type 1

Add, subtract, scalar multiply

Operate entry-by-entry.

Example 1
Compute (1234)+(5678)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}. (A) (681012)\begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} (B) (610812)\begin{pmatrix} 6 & 10 \\ 8 & 12 \end{pmatrix} (C) (5122132)\begin{pmatrix} 5 & 12 \\ 21 & 32 \end{pmatrix} (D) (112214)\begin{pmatrix} 1 & 12 \\ 21 & 4 \end{pmatrix}

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Type 2

Matrix multiplication

Each entry is a row-times-column dot product.

Example 2
Compute (1234)(0110)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. (A) (2143)\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} (B) (1234)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} (C) (0230)\begin{pmatrix} 0 & 2 \\ 3 & 0 \end{pmatrix} (D) (2314)\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}

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Type 3

Matrix-vector product

Av is computed by dotting each row of A with v.

Example 3
Compute (2103)(45)\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ 5 \end{pmatrix}. (A) (1315)\begin{pmatrix} 13 \\ 15 \end{pmatrix} (B) (815)\begin{pmatrix} 8 \\ 15 \end{pmatrix} (C) (95)\begin{pmatrix} 9 \\ 5 \end{pmatrix} (D) (415)\begin{pmatrix} 4 \\ 15 \end{pmatrix}

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