Matrices - ACT Math

3 rows and 2 columns. The first number is rows, second is columns.

A rectangular array of numbers arranged in rows and columns. Numbers organized in a grid format for mathematical operations.

Typically with uppercase letters, e.g., $A$, $B$, $C$. Standard mathematical convention for matrix notation.

The dimensions of a matrix, given as rows $\times$ columns. Specifies the size and shape of the matrix.

A matrix with the same number of rows and columns. Equal dimensions create a special matrix type.

A matrix with a single row. Also called a row vector in linear algebra.

A matrix with a single column. Also called a column vector in linear algebra.

A matrix where all elements are zero. Acts as the additive identity in matrix operations.

A square matrix where all off-diagonal elements are zero. Non-zero elements only appear along the main diagonal.

A square matrix with 1s on the diagonal and 0s elsewhere. Acts as the multiplicative identity in matrix operations.

Switching the rows and columns of a matrix. Rows become columns and columns become rows.

A scalar value that is a function of a square matrix. Measures certain properties of square matrices.

$ad - bc$. Formula for $2 \times 2$ matrix determinant calculation.

A square matrix with a non-zero determinant. Non-zero determinant means the matrix has an inverse.

A matrix that, when multiplied with the original, results in the identity matrix. Denoted as $A^{-1}$ where $A \cdot A^{-1} = I$.

Adding corresponding elements of two matrices of the same order. Both matrices must have identical dimensions.

$\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$. Each row becomes a column in the transposed matrix.

$\begin{bmatrix} 2 & 4 \\ 6 & 8 \\ \end{bmatrix}$. Each element multiplied by $2$: $ (2 \cdot 1, 2 \cdot 2, 2 \cdot 3, 2 \cdot 4) $

No. Not symmetric because $a_{12} \neq a_{21}$ (2 ≠ 3).

Multiplying each element of a matrix by a scalar. Multiply every element by the same scalar value.

Dot product of rows and columns from two matrices. Row elements multiplied by column elements, then summed.

Number of columns in the first matrix must equal the number of rows in the second. Inner dimensions must match for multiplication to work.

A square matrix whose transpose is also its inverse. Special property where $A^T = A^{-1}$.

The sum of the diagonal elements of a square matrix. Add all diagonal elements from top-left to bottom-right.

A square matrix with a determinant of zero. Cannot be inverted due to zero determinant.

Matrix multiplication. Order matters in matrix multiplication: $AB \neq BA$.

Leaves any matrix unchanged when multiplied. Multiplicative identity property: $AI = IA = A$.

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

Subtracting corresponding elements of two matrices of the same order. Subtract element-wise from corresponding positions.