Answer
Verified
399.3k+ views
Hint: First find the value of ${A^2}$ by taking the product of matrix A with itself and then use it to find the value of ${A^4}$ that is the required result.
Complete step by step solution:
We have given a matrix having the form:
$A = \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
The goal is to find the value of ${A^4}$.
We can express ${A^4}$ as ${A^2} \times {A^2}$, so first, we need to find the value of ${A^2}$.
The value of ${A^2}$ is given as the product of a matrix $A$ with itself. That is,
${A^2} = A \times A$
Substitute the value of the matrix given in the problem:
${A^2} = \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
Find the product of matrices,
${A^2} = \left[ {\begin{array}{*{20}{c}}
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3} \\
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3} \\
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}
\end{array}} \right]$
\[{A^2} = \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right]\]
Now, we have the value of${A^2}$. Use this value to find the value of${A^4}$.
${A^4} = {A^2} \times {A^2}$
Substitute the value of the matrix${A^2}$in the above equation:
${A^4} = \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right]$
Find the product of the above matrix:
${A^4} = \left[ {\begin{array}{*{20}{c}}
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27} \\
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27} \\
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}
\end{array}} \right]$
${A^4} = \left[ {\begin{array}{*{20}{c}}
{3 \times 729}&{3 \times 729}&{3 \times 729} \\
{3 \times 729}&{3 \times 729}&{3 \times 729} \\
{3 \times 729}&{3 \times 729}&{3 \times 729}
\end{array}} \right]$
Take out $729$as a common factor from all the elements.
${A^4} = 729\left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
${A^4} = 729A$
We have obtained the value of ${A^4}$ as$729A$.
Therefore, the option (d) is correct.
Note: Before multiplying the matrix, we have to check, that is it possible to multiply the matrices.
We can say that the matrices are possible to multiply if the number of columns in the first matrix is equal to the number of rows of the other matrix. If this condition does not hold then it is not possible to multiply the matrices.
We can see that we have a matrix $A$ of dimension $3 \times 3$, so its multiplication with itself is possible because the number of rows in the matrix is equal to the number of columns in the matrix.
Complete step by step solution:
We have given a matrix having the form:
$A = \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
The goal is to find the value of ${A^4}$.
We can express ${A^4}$ as ${A^2} \times {A^2}$, so first, we need to find the value of ${A^2}$.
The value of ${A^2}$ is given as the product of a matrix $A$ with itself. That is,
${A^2} = A \times A$
Substitute the value of the matrix given in the problem:
${A^2} = \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
Find the product of matrices,
${A^2} = \left[ {\begin{array}{*{20}{c}}
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3} \\
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3} \\
{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}&{3 \times 3 + 3 \times 3 + 3 \times 3}
\end{array}} \right]$
\[{A^2} = \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right]\]
Now, we have the value of${A^2}$. Use this value to find the value of${A^4}$.
${A^4} = {A^2} \times {A^2}$
Substitute the value of the matrix${A^2}$in the above equation:
${A^4} = \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right] \times \left[ {\begin{array}{*{20}{c}}
{27}&{27}&{27} \\
{27}&{27}&{27} \\
{27}&{27}&{27}
\end{array}} \right]$
Find the product of the above matrix:
${A^4} = \left[ {\begin{array}{*{20}{c}}
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27} \\
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27} \\
{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}&{27 \times 27 + 27 \times 27 + 27 \times 27}
\end{array}} \right]$
${A^4} = \left[ {\begin{array}{*{20}{c}}
{3 \times 729}&{3 \times 729}&{3 \times 729} \\
{3 \times 729}&{3 \times 729}&{3 \times 729} \\
{3 \times 729}&{3 \times 729}&{3 \times 729}
\end{array}} \right]$
Take out $729$as a common factor from all the elements.
${A^4} = 729\left[ {\begin{array}{*{20}{c}}
3&3&3 \\
3&3&3 \\
3&3&3
\end{array}} \right]$
${A^4} = 729A$
We have obtained the value of ${A^4}$ as$729A$.
Therefore, the option (d) is correct.
Note: Before multiplying the matrix, we have to check, that is it possible to multiply the matrices.
We can say that the matrices are possible to multiply if the number of columns in the first matrix is equal to the number of rows of the other matrix. If this condition does not hold then it is not possible to multiply the matrices.
We can see that we have a matrix $A$ of dimension $3 \times 3$, so its multiplication with itself is possible because the number of rows in the matrix is equal to the number of columns in the matrix.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE