5I.2 - Square matrices
5I.2 - Content video: Square matrices
This video covers the theory and several examples relating to square matrices.
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A square matrix contains the same number of rows and columns. The order of any square matrix can be written as \(m \times m\). Examples of square matrices include: \[\begin{bmatrix}
3
\end{bmatrix}
\begin{bmatrix}
2 & 4\\
1 & -2
\end{bmatrix}
\begin{bmatrix}
8 & 2 & -2\\
-1 & 3 & 5\\
7 & 4 & 9
\end{bmatrix}\] Important notes for square matrices:
3
\end{bmatrix}
\begin{bmatrix}
2 & 4\\
1 & -2
\end{bmatrix}
\begin{bmatrix}
8 & 2 & -2\\
-1 & 3 & 5\\
7 & 4 & 9
\end{bmatrix}\] Important notes for square matrices:
- Only square matrices can be squared (\(A^2\)).
- Only square matrices have a multiplicative inverse (\(A^{-1}\)).
The identity matrix:
The identity matrix is a square matrix whose entries in the leading diagonal are all , all other elements are zeros. \[I_1=\begin{bmatrix}
1
\end{bmatrix},
I_2=\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix},
I_3=\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\] For the identity matrix, \(I\), and another matrix, \(A\), we observe: \(AI=A=IA\)
The identity matrix is a square matrix whose entries in the leading diagonal are all , all other elements are zeros. \[I_1=\begin{bmatrix}
1
\end{bmatrix},
I_2=\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix},
I_3=\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\] For the identity matrix, \(I\), and another matrix, \(A\), we observe: \(AI=A=IA\)
The multiplicative inverse:
The multiplicative inverse of \(A\) is denoted by \(A^{-1}\) (this is not equal to \(\frac{1}{A}\)). The multiplicative inverse, \(A^{-1}\), is defined to be the matrix such that \(AA^{-1}=I=A^{-1}A\). For the matrix \(A=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\), the multiplicative inverse is given by: \[A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}
d & -b\\
-c & a
\end{bmatrix}\] The term \(ad-bc\) is known as as the discriminant and can be represented as \(det(A)\) or \(|A|\). If \(det(A)=0\) then the inverse matrix, \(A^{-1}\), does not exist.
The multiplicative inverse of \(A\) is denoted by \(A^{-1}\) (this is not equal to \(\frac{1}{A}\)). The multiplicative inverse, \(A^{-1}\), is defined to be the matrix such that \(AA^{-1}=I=A^{-1}A\). For the matrix \(A=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\), the multiplicative inverse is given by: \[A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}
d & -b\\
-c & a
\end{bmatrix}\] The term \(ad-bc\) is known as as the discriminant and can be represented as \(det(A)\) or \(|A|\). If \(det(A)=0\) then the inverse matrix, \(A^{-1}\), does not exist.
5I.2 - Example 1: Calculating the inverse of a matrix (CAS)
Let \(A=\begin{bmatrix}
2 & 3\\ 3 & 5 \end{bmatrix}\)
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5I.2 - Example 1: Video solution
5I.2 - Example 1: Practice
Question 1: ABC Question 2: ABC 5I.2 - Example 1: Solutions
Question 1: ABC Question 2: ABC |
5I.2 - Example 2: Calculating the inverse of a matrix
Let \(C=\begin{bmatrix}
p & 3\\ 12 & p \end{bmatrix}\), where \(p\in \mathbb{R}\)
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5I.2 - Example 2: Video solution
5I.2 - Example 2: Practice
Question 1: ABC Question 2: ABC 5I.2 - Example 2: Solutions
Question 1: ABC Question 2: ABC |