
If \[A=\left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\], find the determinant of the matrix \[{{A}^{2}}-2A\].
Answer
412.5k+ views
Hint: Here in this question, we need to find the determinant of the matrix \[{{A}^{2}}-2A\]. Before solving this, we need to look at the definition of matrix. After that, we will consider the given data and given expression, firstly we are going to take the determinant to the \[{{A}^{2}}-2A\], then evaluate the answer.
Complete step by step answer:
Matrix is defined as the rectangular arrangement of numbers (real or complex) which may be represented as
\[\left( \begin{matrix}
{{a}_{11}} & \ldots & {{a}_{1n}} \\
\vdots & \ddots & \vdots \\
{{a}_{m1}} & \cdots & {{a}_{mn}} \\
\end{matrix} \right)\]
Matrix is enclosed by \[\left( {} \right)\] or \[\left[ {} \right]\].
Compact from the above matrix is represented by \[{{\left[ {{a}_{ij}} \right]}_{m\times n}}\]or \[A=\left[ {{a}_{ij}} \right]\].
Let us solve the given question,
Given data \[A=\left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\],
Given expression, \[{{A}^{2}}-2A\]
To find determinant \[{{A}^{2}}-2A\]
Now,
\[\left| {{A}^{2}}-2A \right|=\left| A\left( A-2I \right) \right|\]
(Taking A common on Right-hand-side)
Writing the determinants separately, on the basis of \[\left( \left| AB \right|=\left| A \right|\left| B \right| \right)\], then we get
\[\Rightarrow \left| A \right|\left| A-2I \right|\]
We are going to substituting the matrix \[A=\left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\] and identity matrix \[I=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]\] is an identity matrix same as \[2\times 2\] on above expression \[{{A}^{2}}-2A\],
\[\Rightarrow \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]\]
We will multiply the first two matrices first and then we will multiply the resultant to remaining matrix, then we get
\[\Rightarrow \left[ \left( 1\times 1-2\times 3 \right) \right]\times \left| \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \right|\]
Above matrix is obtained by \[2\times 2\]matrix multiplication,
\[\Rightarrow \left( 1-6 \right)\times \left| \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \right|\]
On solving,
\[\Rightarrow -5\times \left| \left[ \begin{matrix}
1-2 & 3-0 \\
2-0 & 1-2 \\
\end{matrix} \right] \right|\]
On further evaluation,
\[\Rightarrow -5\times \left| \left[ \begin{matrix}
-1 & 3 \\
2 & -1 \\
\end{matrix} \right] \right|\]
Finding the determinant of the above matrix,
\[\Rightarrow -5\times \left( \left( -1\times -1 \right)-2\times 3 \right)\]
Multiplying the above terms,
\[\Rightarrow -5\times \left( 1-6 \right)\]
\[\Rightarrow -5\times -5\]
Therefore, \[{{A}^{2}}-2A=25\].
Note: It is important to note that when we consider two matrices to be equal then in order to hold the equality every corresponding element in both the matrices should be equal. For matrix multiplication, the number of columns present in the first matrix should be equal to the number of rows present in the second matrix.
Complete step by step answer:
Matrix is defined as the rectangular arrangement of numbers (real or complex) which may be represented as
\[\left( \begin{matrix}
{{a}_{11}} & \ldots & {{a}_{1n}} \\
\vdots & \ddots & \vdots \\
{{a}_{m1}} & \cdots & {{a}_{mn}} \\
\end{matrix} \right)\]
Matrix is enclosed by \[\left( {} \right)\] or \[\left[ {} \right]\].
Compact from the above matrix is represented by \[{{\left[ {{a}_{ij}} \right]}_{m\times n}}\]or \[A=\left[ {{a}_{ij}} \right]\].
Let us solve the given question,
Given data \[A=\left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\],
Given expression, \[{{A}^{2}}-2A\]
To find determinant \[{{A}^{2}}-2A\]
Now,
\[\left| {{A}^{2}}-2A \right|=\left| A\left( A-2I \right) \right|\]
(Taking A common on Right-hand-side)
Writing the determinants separately, on the basis of \[\left( \left| AB \right|=\left| A \right|\left| B \right| \right)\], then we get
\[\Rightarrow \left| A \right|\left| A-2I \right|\]
We are going to substituting the matrix \[A=\left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\] and identity matrix \[I=\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]\] is an identity matrix same as \[2\times 2\] on above expression \[{{A}^{2}}-2A\],
\[\Rightarrow \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]\times \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right]\]
We will multiply the first two matrices first and then we will multiply the resultant to remaining matrix, then we get
\[\Rightarrow \left[ \left( 1\times 1-2\times 3 \right) \right]\times \left| \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \right|\]
Above matrix is obtained by \[2\times 2\]matrix multiplication,
\[\Rightarrow \left( 1-6 \right)\times \left| \left[ \begin{matrix}
1 & 3 \\
2 & 1 \\
\end{matrix} \right]-2\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
\end{matrix} \right] \right|\]
On solving,
\[\Rightarrow -5\times \left| \left[ \begin{matrix}
1-2 & 3-0 \\
2-0 & 1-2 \\
\end{matrix} \right] \right|\]
On further evaluation,
\[\Rightarrow -5\times \left| \left[ \begin{matrix}
-1 & 3 \\
2 & -1 \\
\end{matrix} \right] \right|\]
Finding the determinant of the above matrix,
\[\Rightarrow -5\times \left( \left( -1\times -1 \right)-2\times 3 \right)\]
Multiplying the above terms,
\[\Rightarrow -5\times \left( 1-6 \right)\]
\[\Rightarrow -5\times -5\]
Therefore, \[{{A}^{2}}-2A=25\].
Note: It is important to note that when we consider two matrices to be equal then in order to hold the equality every corresponding element in both the matrices should be equal. For matrix multiplication, the number of columns present in the first matrix should be equal to the number of rows present in the second matrix.
Recently Updated Pages
Master Class 12 English: Engaging Questions & Answers for Success

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 Social Science: Engaging Questions & Answers for Success

Master Class 12 Chemistry: Engaging Questions & Answers for Success

Class 12 Question and Answer - Your Ultimate Solutions Guide

Master Class 12 Economics: Engaging Questions & Answers for Success

Trending doubts
Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is a transformer Explain the principle construction class 12 physics CBSE

Draw a labelled sketch of the human eye class 12 physics CBSE

What are the major means of transport Explain each class 12 social science CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?
