Matrices
In this course, you will learn how matrices are used to organise information clearly and solve mathematical problems step by step.
You will be introduced to matrix notation, rows and columns, and how to read and interpret values from a matrix.
You will also learn key operations such as addition, subtraction, scalar multiplication, and matrix multiplication of matrices.
The course will also guide you on how to apply matrices to solve problem sums involving the use of Matrices.
Related Lessons:
What Is The Order Of Matrix?
The order tells us the number of rows and columns in a matrix, and it helps us describe the size and structure of the information being presented.
- If a matrix has 2 rows and 3 columns, the order of the matrix is 2×3
- If a matrix has 5 rows and 4 columns, the order of the matrix is 5×4
Add & Subtract Matrices
Matrices can only be added or subtracted when they have the same order, as each value must be matched with the corresponding position in the other matrix. For example:
\[
\begin{align*}
\left(\begin{matrix}1&2\\-3&4\end{matrix}\right)+\left(\begin{matrix}2&4\\1&3\end{matrix}\right)&=\left(\begin{matrix}1+2&2+4\\-3+1&4+3\end{matrix}\right)\\
&=\left(\begin{matrix}3&6\\-2&7\end{matrix}\right)
\end{align*}
\]
A scalar is simply a single number. When a scalar multiplied a matrix, each element in the matrix must be multiplied by that number. For example:
\[
\begin{align*}
4\left(\begin{matrix}3&6\\-2&7\end{matrix}\right)&=\left(\begin{matrix}4\times3&4\times6\\4\times\left(-2\right)&4\times7\end{matrix}\right)\\
&=\left(\begin{matrix}12&24\\-8&28\end{matrix}\right)
\end{align*}
\]
Multiplication Of Matrices
Two matrices can be multiplied only if the number of columns in the first matrix matches the number of rows in the second matrix. For example:
- 2×3 matrix cannot multiple with a 2×3 since the first matrix column value is 3 and the second matrix row is 2.
- 2×3 matrix can multiple with a 3×4 matrix since first matrix column is 3 and the second matrix row is also 3. The resultant matrix will be 2×4, taking the first matrix row value and the second matrix column value.
The video lesson will guide you step by step through the process of multiplying rows by columns to find each value in the new matrix systematically, helping you reduce careless mistakes and improve accuracy.
Example 1
- Express \(3P-Q\) as a single matrix
- Find \(P^2\)
- \(PQ\)
Example 2
Find the values of a and b such that \(2\left(\begin{matrix}a+2\\2b-3\end{matrix}\right)+\left(\begin{matrix}3b-4\\6-a\end{matrix}\right)=\left(\begin{matrix}18\\13\end{matrix}\right)\)
Example 3
Evaluate \(\left(\begin{matrix}0&1\\-3&3\end{matrix}\right)\left(\begin{matrix}2\\3\end{matrix}\right)\)
Example 4
Find the matrix M such that \(3M+\left(\begin{matrix}1&0\\1&-1\end{matrix}\right)=\left(\begin{matrix}4&3\\7&-7\end{matrix}\right)\)
Example 5
- Represent the data in the table using a 4×3 matrix M.
- Given that \(N=M\left(\begin{matrix}1\\2\\10\end{matrix}\right)\), evaluate N and explain what the element(s) of N represents.
- Given that \(P=\frac{1}{4}\left(\begin{matrix}1&1&1\end{matrix}\right)N\), evaluate P and explain what the elements in P represents.