2 Matrices and Matrix Operations Free Online FlashCards

What do the subscripts in $a_{ij}$ identify?

An an entry’s first subscript gives its row, and its second subscript gives its column. Thus, in a matrix, the entry in row 2, column 3 is written as $a_{23}$.

What is an augmented matrix?

An augmented matrix combines a system’s coefficient matrix with its constants column, separated by a vertical bar. For $A\mathbf{x}=\mathbf{b}$, it is written $[A\mid\mathbf{b}]$.

When can two matrices be added?

Matrices can be added or subtracted only when they have the same dimensions. The operation is performed entry by entry.

What does scalar multiplication do to a matrix?

A scalar multiple multiplies every entry of a matrix by the same scalar. For example, multiplying by $3$ triples every entry.

When is the product $AB$ defined?

If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is defined and has size $m\times p$.

How is an entry of a matrix product calculated?

Each entry of $AB$ is the dot product of a row of $A$ with a column of $B$: $(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$.

Is matrix multiplication commutative?

Generally, $AB\ne BA$. In addition, one product may be defined while the other is not, because the dimension requirements can differ.

What does the transpose of a matrix do?

The transpose exchanges rows and columns, so an $m\times n$ matrix becomes an $n\times m$ matrix. Its entries satisfy $(A^T)_{ij}=a_{ji}$.

What is the transpose-of-a-product identity?

The transpose of a product reverses the order: $(AB)^T=B^TA^T$. The reversal preserves dimension compatibility.

What is the inverse formula for a $2\times2$ matrix?

For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, if $ad-bc\ne0$, then $A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$.

How can row reduction determine whether a square matrix is invertible?

A square matrix is invertible exactly when its rows can be reduced to the identity matrix, equivalently when it has a pivot in every row and column.

How does Gauss–Jordan elimination find an inverse?

To find $A^{-1}$, row-reduce the augmented matrix $[A\mid I_n]$ until it becomes $[I_n\mid A^{-1}]$.