Lesson 29 - Quick Lesson on Matrices

Before we get into solving linear homogeneous systems using matrix methods, let's introduce some quick linear algebra.

Matrices

A matrix with $m$ rows and $n$ columns is denoted by $A = [a_{i j}]_{m \times n}$

A = | \begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m_{1}} & a_{m_{2}} & \dots & a_{m n} \end{matrix} |

A matrix is called a square matrix if the dimensions $m = n$ .

Two matrices can be added or subtracted if they have the same number of rows and columns.
If $A$ is a matrix and $B$ is a matrix and both dimensions are equal, then

A + B = | \begin{matrix} a_{11} + b_{11} & a_{12} + b_{12} & \dots & a_{1 n} + b_{1 n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \dots & a_{2 n} + b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 m} + b_{1 m} & a_{2 m} + b_{2 m} & \dots & a_{m n} \end{matrix} |

In order to perform matrix multiplication, the inner dimensions of the matrices must agree. Meaning, if $A$ has dimensions $2 \times 1$ and B has dimensions $1 \times 2$ , then these matrices can be multiplied together because the inner dimensions agree. Technically, there are different types of matrix multiplications, such as matrix-matrix multiplication, scalar-matrix multiplication, scalar-vector multiplication, and others. To learn more, visit An Introduction to Linear Algebra.

Consider the matrices:

A = | \begin{matrix} 1 & 2 & 3 \\ - 1 & 5 & 3 \\ 3 & 2 & - 2 \end{matrix} |

B = | \begin{matrix} 2 \\ 3 \\ - 1 \end{matrix} |

Let's perform matrix multiplication. We perform this by multiplying each row in $A$ by each element in the single row of $B$ .

A \times B = | \begin{matrix} (1 \cdot 2) & (2 \cdot 3) & (3 \cdot - 1) \\ (- 1 \cdot 2) & (5 \cdot 3) & (3 \cdot - 1) \\ (3 \cdot 2) & (2 \cdot 3) & (- 2 \cdot - 1) \end{matrix} |

A \times B = | \begin{matrix} 2 & 6 & - 3 \\ - 2 & 15 & - 3 \\ 6 & 6 & 2 \end{matrix} |

One additional important step. Remember when we talked about the inner dimensions? The dimensions here are $3 \times 3$ for matrix $A$ and $3 \times 1$ for matrix $B$ . When we talk about multiplying matrices, the inner dimensions will combine and this will give us the new dimensions for our product. So when we multiply a matrix with dimensions $a \times b$ with a matrix with dimensions $b \times c,$ the new dimensions become $a \times c$ .

In our example above, the dimensions for the product matrix will be $3 \times 1$ . We will add each row in the matrix $A \times B$ in order to get to the correct dimensions, and this will give us the correct product matrix.

A \times B = | \begin{matrix} 2 + 6 + (- 3) \\ - 2 + 15 + (- 3) \\ 6 + 6 + 2 \end{matrix} |

A \times B = | \begin{matrix} 5 \\ 10 \\ 14 \end{matrix} |

We can also represent a system of linear equations in matrix form:

Consider the linear system: {\begin{cases} 3 x_{1} + 4 x_{2} - x_{3} = - 1 \\ 2 x_{1} - x_{2} + x_{3} = 1 \\ x_{1} - x_{2} - x_{3} = 2 \end{cases}

We can extract the coefficients of the linear system in what we call a coefficient matrix. Then, the leftover variables can be input into a $3 \times 1$ vector.

| \begin{matrix} 3 & 4 & - 1 \\ 2 & - 1 & 1 \\ 1 & - 1 & - 1 \end{matrix} | | \begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix} | = | \begin{matrix} - 1 \\ 1 \\ 2 \end{matrix} |

Try For Yourself

A = | \begin{matrix} 1 & 2 & 3 \\ 5 & - 2 & 2 \\ 1 & 2 & 2 \end{matrix} | B = | \begin{matrix} 1 & 3 & 5 \\ - 1 & 2 & 1 \\ 2 & 0 & - 1 \end{matrix} | C = | \begin{matrix} 1 \\ - 3 \\ 2 \end{matrix} |

Given matrices $A$ , $B$ , and $C$ , find the following:

$A \times C$
$B \times C$
$A + B$
$A - B$

Next Lesson: Lesson 30 - Matrix Methods for Linear Homogeneous Systems

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