Answer
398.4k+ views
Hint: Now we are given with a square matrix of order 2. Now we know that the determinant of a order 2 matrix is given by $\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)$ hence using this we will get the first equation. Now again calculate the determinant of 3A and simplify. Now substituting the value from the first equation we will get the value of determinant of 3A.
Complete step by step answer:
Now let us first understand matrices and determinants.
Now matrices are nothing but rectangular arrays consisting of rows and columns.
Hence we have a matrix A with m rows and n columns as $A=\left[ \begin{matrix}
{{a}_{1}} & ... & {{a}_{m}} \\
\vdots & \ddots & \vdots \\
{{a}_{n}} & ... & {{a}_{mn}} \\
\end{matrix} \right]$ .
Now if the matrix has an equal number of rows and columns then we say the matrix is a square matrix.
Hence if a matrix has n rows and n columns the order of the matrix is called n or n × n.
For a square matrix we can define determinant.
Determinant is calculated in a very particular manner.
For matrix of order 2 the determinant is given by,
$\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)$ .
Now let us consider a 2 × 2 matrix A such that the determinant of A is – 5.
Hence we have, $\left| A \right|=\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)=-5........\left( 1 \right)$
Now consider the matrix 3A.
$3A=\left[ \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right]$
Now let us calculate the determinant of this matrix.
$\begin{align}
& \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=\left( 3{{a}_{11}}3{{a}_{22}}-3{{a}_{21}}3{{a}_{12}} \right) \\
& \Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=\left( 9{{a}_{11}}{{a}_{22}}-9{{a}_{21}}{{a}_{12}} \right) \\
& \Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=9\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right) \\
\end{align}$
Now from equation (1) we get,
$\Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=9\left( -5 \right)=-45$
Hence we get $\left| 3A \right|=-45$.
Note: Now note that we can directly find the value of the determinant of matrix 3A. the determinant is given by ${{3}^{n}}\left| A \right|$ where n is the order of the matrix. In general we have that if r is a scalar and A is a matrix of order n then $\left| rA \right|={{r}^{n}}\left| A \right|$ .
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)$ hence using this we will get the first equation. Now again calculate the determinant of 3A and simplify. Now substituting the value from the first equation we will get the value of determinant of 3A.
Complete step by step answer:
Now let us first understand matrices and determinants.
Now matrices are nothing but rectangular arrays consisting of rows and columns.
Hence we have a matrix A with m rows and n columns as $A=\left[ \begin{matrix}
{{a}_{1}} & ... & {{a}_{m}} \\
\vdots & \ddots & \vdots \\
{{a}_{n}} & ... & {{a}_{mn}} \\
\end{matrix} \right]$ .
Now if the matrix has an equal number of rows and columns then we say the matrix is a square matrix.
Hence if a matrix has n rows and n columns the order of the matrix is called n or n × n.
For a square matrix we can define determinant.
Determinant is calculated in a very particular manner.
For matrix of order 2 the determinant is given by,
$\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)$ .
Now let us consider a 2 × 2 matrix A such that the determinant of A is – 5.
Hence we have, $\left| A \right|=\left| \begin{matrix}
{{a}_{11}} & {{a}_{12}} \\
{{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right|=\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right)=-5........\left( 1 \right)$
Now consider the matrix 3A.
$3A=\left[ \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right]$
Now let us calculate the determinant of this matrix.
$\begin{align}
& \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=\left( 3{{a}_{11}}3{{a}_{22}}-3{{a}_{21}}3{{a}_{12}} \right) \\
& \Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=\left( 9{{a}_{11}}{{a}_{22}}-9{{a}_{21}}{{a}_{12}} \right) \\
& \Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=9\left( {{a}_{11}}{{a}_{22}}-{{a}_{21}}{{a}_{12}} \right) \\
\end{align}$
Now from equation (1) we get,
$\Rightarrow \left| \begin{matrix}
3{{a}_{11}} & 3{{a}_{12}} \\
3{{a}_{21}} & 3{{a}_{22}} \\
\end{matrix} \right|=9\left( -5 \right)=-45$
Hence we get $\left| 3A \right|=-45$.
Note: Now note that we can directly find the value of the determinant of matrix 3A. the determinant is given by ${{3}^{n}}\left| A \right|$ where n is the order of the matrix. In general we have that if r is a scalar and A is a matrix of order n then $\left| rA \right|={{r}^{n}}\left| A \right|$ .
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)