5I.1 - Introduction to matrices
5I.1 - Content video: Introduction to matrices
This video covers the theory and several examples relating to matrices.
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A matrix (plural matrices) is a rectangular array of numbers organised into rows and columns.
- The numbers within a matrix are referred to as elements.
- The order of a matrix is defined by the number of rows and columns:
Order = (numbers of rows) \(\times\) (number of columns)
Generally, capital letters (\(A,B,C,...\)) are used to represent matrices. Consider the matrix shown below \[A=\begin{bmatrix}
1 & 2 & 4\\
0 & -7 & 3
\end{bmatrix}\]
1 & 2 & 4\\
0 & -7 & 3
\end{bmatrix}\]
- This matrix has 2 rows and 3 columns; therefore, we say that it is a rectangular matrix with order 2 by 3 (written as: \(2\times 3\)).
- When referring to a specific element we can use the following notation \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number. For an \(m\times n\) matrix, the \(a_{ij}\) notation is as follows: \[A=\begin{bmatrix}
a_{11} & a_{12} & ... & a_{1n}\\
a_{21} & a_{22} & ... & a_{2n}\\
\vdots & \vdots & & \vdots\\
a_{m1} & a_{m2} & ... & a_{mn}
\end{bmatrix}=\begin{bmatrix}
a_{ij}
\end{bmatrix}\] - Two matrices, \(A\) and \(B\), are equivalent if they have the same order and all corresponding elements are equal.
Row, column and square matrices:
- A row matrix (or row vector) contains only one row. The following is a \(1\times 3\) row matrix: \(\begin{bmatrix}
a & b & c
\end{bmatrix}\) - A column matrix (or column vector) contains only one column. The following is a \(3\times 1\) column matrix: \(\begin{bmatrix}
a\\
b\\
c
\end{bmatrix}\) - A square matrix contains the same amount of rows and columns. The following is a \(2\times 2\) square matrix: \(\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\)
Addition of matrices:
To be able to add two matrices, they must have the same order (\(m\times n\)).
To be able to add two matrices, they must have the same order (\(m\times n\)).
- If the matrices are the same order, simply add the corresponding elements together and put the sum in the resultant matrix. Corresponding elements have the same \(x_{ij}\) location.
- If addition is defined we find that \(A+B=B+A\).
Subtraction of matrices:
To be able to subtract two matrices, they must have the same order (\(m\times n\)).
To be able to subtract two matrices, they must have the same order (\(m\times n\)).
- If the matrices are the same order, simply subtract the corresponding elements and put the difference in the resultant matrix. Corresponding elements have the same \(x_{ij}\) location.
- If subtraction is defined we find that \(A+B\neq B+A\) with some exceptions.
Scalar multiplication of matrices:
When multiplying a matrix by a scalar, multiply every entry by the scalar.
When multiplying a matrix by a scalar, multiply every entry by the scalar.
- When a number is placed out the front of a matrix it is known as a scalar. The scalar of a matrix behaves in the same way the number \(2\) behaves in \(2A\). That is, it multiplies \(A\) by \(2\).
- For matrices, it scales each entry by the value of a scalar.
5C.1 - Example 1: Arithmetic involving matrices
If \(A=\begin{bmatrix}
1 & 3\\ 8 & 2 \end{bmatrix}\) and \(B=\begin{bmatrix} 2 & 4\\ -1 & -3 \end{bmatrix}\), find \(C\) such that \(C+B=5A\) |
5C.1 - Example 1: Video solution
5C.1 - Example 1: Practice
Question 1: ABC Question 2: ABC 5C.1 - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Matrix multiplication:
5I.1 - Content video: Matrix multiplication
This video covers the theory and several examples relating to the multiplication of matrices.
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Two matrices can only be multiplied together if the number of columns in the first matrix is equal to the number of rows in the second. In general, multiplication is defined when the orders of the matrices follow the pattern: \[(a \times b)\times(b \times c)\]
1 & 4 & 9\\
-6 & -3 & 0
\end{bmatrix} \times
\begin{bmatrix}
4 & 5\\
1 & -2\\
7 & 1
\end{bmatrix}=
\begin{bmatrix}
71 & 6\\
-21 & -24
\end{bmatrix}\]
- The resultant matrix will have order \(a \times c\).
- If multiplication is defined, we find that \(AB \neq BA\) with some exceptions.
1 & 4 & 9\\
-6 & -3 & 0
\end{bmatrix} \times
\begin{bmatrix}
4 & 5\\
1 & -2\\
7 & 1
\end{bmatrix}=
\begin{bmatrix}
71 & 6\\
-21 & -24
\end{bmatrix}\]
- Multiplication is defined because the order follows the pattern described above: \((2\times 3)\times (3 \times 2)\).
- The resultant matrix has order \(2\times 2\).
■ To multiply matrices we take the \(i^{th}\) row of the first matrix and multiply the \(j^{th}\) column of the second matrix. The result of this process occupies that \(x_{ij}\) position of the resultant matrix.
This process is best illustrated with several examples which are included in the content video for this section.