1.3 Matrices and Matrix Operations

<< 1.2 Gaussian Elimination

Definition: Matrix

A matrix a rectangular array of numbers. The numbers in the array are called the entries in the matrix.

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Definition: Matrix equality

Two matrices are defined to be equal if they have the same size and their corresponding entries are equal.

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Definition: Matrix operations

Let $A$ and $B$ are matrices of the same size

Then

• Sum $A+B$ is the matrix obtained by adding the entries of $B$ to the corresponding entries of $B$
• Difference $A-B$ is the matrix obtained by subtracting the entries of $B$ from the corresponding entries of $A$.
• Matrices of different sizes cannot be added or subtracted.

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Definition: Matrix scalar multiplication

Let

• $A$ : Matrix
• $c$ : Scalar

Then

• Product $cA$ is the matrix obtained by multiplying entry of $A$ by $c$
• We say $cA$ a scalar multiple of $A$

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Definition: Matrix product

Let

• $A$ : $m\times r$ matrix
• $B$ : $r\times n$ matrix

Then the product $AB$ is a $m\times n$ matrix, whose entries are determined as follows:

1. To find the entry in row $i$ and column $j$ of $AB$, single out row $i$ from the matrix $A$ and column $j$ from the matrix $B$.
2. Multiply the corresponding entries from the row and column together, and then add up the resulting products.

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Exercise

Matrix multiplication

Consider the matrices

$$A=\left[\begin{array}{ccc}1& 2& 4\\ 2& 6& 0\end{array}\right],B=\left[\begin{array}{cccc}4& 1& 4& 3\\ 0& -1& 3& 1\\ 2& 7& 5& 2\end{array}\right]$$

Since $A$ is a $2\times 3$ matrix, and $B$ is a $3\times 4$ matrix, the product $AB$ is a $2\times 4$ matrix.

The process of matrix multiplication is illlustrated below:

Entry of $(AB{)}_{23}$ is:

$$\left[\begin{array}{ccc}1& 2& 4\\ 2& 6& 0\end{array}\right]\left[\begin{array}{cccc}4& 1& 4& 3\\ 0& -1& 3& 1\\ 2& 7& 5& 2\end{array}\right]=\left[\begin{array}{cccc}\square & \square & \square & \square \\ \square & \square & 26& \square \end{array}\right]$$

Entry of $(AB{)}_{14}$ is:

$$\left[\begin{array}{ccc}1& 2& 4\\ 2& 6& 0\end{array}\right]\left[\begin{array}{cccc}4& 1& 4& 3\\ 0& -1& 3& 1\\ 2& 7& 5& 4\end{array}\right]=\left[\begin{array}{cccc}\square & \square & \square & 13\\ \square & \square & \square & \square \end{array}\right]$$

The computation for the remaining entries are:

$$\begin{array}{rl}(1\cdot 4)+(2\cdot 0)+(4\cdot 2)& =12\\ (1\cdot 1)-(2\cdot 1)+(4\cdot 7)& =27\\ (1\cdot 4)+(2\cdot 3)+(4\cdot 5)& =30\\ (2\cdot 4)+(6\cdot 0)+(0\cdot 2)& =8\\ (2\cdot 1)-(6\cdot 1)+(0\cdot 8)& =-4\\ (2\cdot 3)+(6\cdot 1)+(0\cdot 2)& =12\end{array}$$

Obtaining

$$AB=\left[\begin{array}{cccc}12& 27& 30& 13\\ 8& -4& 26& 12\end{array}\right]$$