Matrices Modeling Contexts - AP Precalculus

Add corresponding elements of the matrices. Element-wise addition: $(A + B){ij} = a{ij} + b_{ij}$.

A matrix obtained by swapping rows and columns. If $A$ has element $a_{ij}$, then $A^T$ has element $a_{ji}$.

A matrix in which all elements are zero. Also called the null matrix, denoted as $O$ or $\mathbf{0}$.

A square matrix with 1s on the diagonal and 0s elsewhere. Denoted as $I_n$ for an $n \times n$ identity matrix.

3 rows and 4 columns. Dimensions specify the matrix structure as rows by columns.

3 rows and 4 columns. Dimensions specify the matrix structure as rows by columns.

Rank is 1. Second row is $3$ times the first row, making them dependent.

Rank is 1. Second row is $3$ times the first row, making them dependent.

The number of linearly independent rows or columns. Also equals the dimension of the row or column space.

Add corresponding elements of the matrices. Element-wise addition: $(A + B){ij} = a{ij} + b_{ij}$.

A matrix with a determinant of zero. Also called a degenerate matrix; has no inverse.

A matrix that, when multiplied by the original, yields the identity matrix. Denoted as $A^{-1}$ where $AA^{-1} = I$.

$ad - bc$. Standard formula for $2 \times 2$ determinant calculation.

Multiply each element by the scalar. Scalar $k$ times matrix $A$ gives $(kA){ij} = ka{ij}$.

Sum of products of row elements and column elements. $(AB){ij} = \sum{k} a_{ik}b_{kj}$ for compatible dimensions.

A rectangular array of numbers arranged in rows and columns. This is the standard mathematical definition of a matrix structure.

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Standard form of the $2 \times 2$ identity matrix $I_2$.

$a_{ij}$. Standard notation where $i$ is row index, $j$ is column index.

Determinant is -14. $\det = (3)(6) - (8)(4) = 18 - 32 = -14$.

A matrix whose transpose is its inverse. Property: $A^T A = I$ or equivalently $A^T = A^{-1}$.

A matrix with a non-zero determinant. Opposite of singular matrix; has an inverse matrix.

The matrix remains unchanged. Identity matrix is the multiplicative identity for matrices.

Number of rows by number of columns. Written as $m \times n$ for $m$ rows and $n$ columns.

$\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$. Multiply each element by the scalar: $2 \times$ each element.

$\begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$. Add corresponding elements: $(1+5, 2+6, 3+7, 4+8)$.

$(AB)C = A(BC)$ for matrices $A$, $B$, and $C$. Matrix multiplication is associative but not commutative.

Commutative property. Generally $AB \neq BA$ for matrix multiplication.

Same dimensions and all corresponding elements are equal. Matrices $A$ and $B$ are equal if $a_{ij} = b_{ij}$ for all $i,j$.

Trace is 5. $\text{tr}(A) = 1 + 4 = 5$ for the diagonal elements.