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Consider the two matrices \[A\] and \[B\] where \[A = \left[ {\begin{array}{*{20}{c}}1&2\\4&3\end{array}} \right]\] ; \[B = \left[ {\begin{array}{*{20}{c}}5\\{ - 3}\end{array}} \right]\] . Let \[n\left( A \right)\] denote the number of elements in \[A\]. When the two matrices \[X\] and \[Y\] are not conformable for multiplication, then \[n\left( {XY} \right) = 0\]
If \[C = \left( {AB} \right)\left( {B'A} \right)\]; \[D = \left( {B'A} \right)\left( {AB} \right)\] , then find the value of
 \[\left[ {\dfrac{{n\left( C \right)\left( {{{\left| D \right|}^2} + n\left( D \right)} \right)}}{{n\left( A \right) - n\left( B \right)}}} \right]\]

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here, we need to find the value of the given expression which includes the multiplication of the matrices. We will first find the transpose of the second matrix. Then we will find the number of elements of the required matrices. We will substitute all the values in the given expression to get the required value.

Complete step-by-step answer:
 It is given that:
 \[A = \left[ {\begin{array}{*{20}{c}}1&2\\4&3\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{c}}5\\{ - 3}\end{array}} \right]\].
We know the number of elements present in any matrix is equal to the product of the number of rows and number of columns of the matrix.
 \[n\left( A \right) = 2 \times 2 = 4\] ………. \[\left( 1 \right)\]
 \[n\left( B \right) = 2 \times 1 = 2\] ………… \[\left( 2 \right)\]
Now, we will find the transpose of the matrix \[B\] which is denoted by \[B'\] which can be calculated by interchanging their rows and columns.
Therefore,
 \[B' = \left[ {\begin{array}{*{20}{c}}5&{ - 3}\end{array}} \right]\]
It is given that \[C = \left( {AB} \right)\left( {B'A} \right)\].
But we need only \[n\left( C \right)\] in the expression, which means the number of elements present in the matrix.
For that, we will calculate the order of the matrix \[C\]
We know that when we multiply two matrices, then the matrix obtained by their product contains a number of rows equal to the number of rows of the first matrix and number of columns equal to the number of columns of the second matrix.
We will use the same concept now.
We know that the order of matrix \[A\] is \[2 \times 2\] and order of matrix \[B\] is \[2 \times 1\]
So, the order of the matrix obtained by their product is equal to \[2 \times 1\].
Therefore, the order of the matrix \[AB\] is equal to \[2 \times 1\].
We know that the order of matrix \[A\] is \[2 \times 2\] and order of matrix \[B'\] is \[1 \times 2\]
So, the order of the matrix obtained by their product i.e. \[B'A\] is equal to \[1 \times 2\].
Therefore, the order of the matrix \[B'A\] is equal to \[1 \times 2\].
Now, we need the order of the matrix \[C\] which is equal to \[\left( {AB} \right)\left( {B'A} \right)\]
As the order of matrix \[AB\] is equal to \[2 \times 1\] and order of matrix \[B'A\] is equal to \[1 \times 2\].
So the order of the matrix obtained by their product i.e. \[\left( {AB} \right)\left( {B'A} \right)\] is equal to \[2 \times 2\].
We know the number of elements present in any matrix is equal to the product of the number of rows and number of columns of the matrix.
Therefore, the number of elements of the matrix \[C\] is equal to \[2 \times 2 = 4\] .
Thus,
 \[n\left( C \right) = 4\] ……….. \[\left( 3 \right)\]
Now, we need to find the matrix \[D\] which is equal to \[\left( {B'A} \right)\left( {AB} \right)\]
For that, we will find the product of \[B'A\].
 \[B'A = {\left[ {\begin{array}{*{20}{c}}5&{ - 3}\end{array}} \right]_{1 \times 2}} \times {\left[ {\begin{array}{*{20}{c}}1&2\\4&3\end{array}} \right]_{2 \times 2}}\]
On multiplying these two matrices, we get
 \[ \Rightarrow B'A = {\left[ {\begin{array}{*{20}{c}}{5 + \left( { - 3} \right) \times 4}&{5 \times 2 + \left( { - 3} \right) \times 3}\end{array}} \right]_{1 \times 2}}\]
On further simplification, we get
 \[ \Rightarrow B'A = {\left[ {\begin{array}{*{20}{c}}{ - 7}&1\end{array}} \right]_{1 \times 2}}\]
Now, we will find the product of \[AB\].
 \[AB = {\left[ {\begin{array}{*{20}{c}}1&2\\4&3\end{array}} \right]_{2 \times 2}} \times {\left[ {\begin{array}{*{20}{c}}5\\{ - 3}\end{array}} \right]_{2 \times 1}}\]
On multiplying these two matrices, we get
 \[ \Rightarrow AB = {\left[ {\begin{array}{*{20}{c}}{5 + 2 \times - 3}\\{4 \times 5 + 3 \times - 3}\end{array}} \right]_{2 \times 1}}\]
On further simplification, we get
 \[ \Rightarrow AB = {\left[ {\begin{array}{*{20}{c}}{ - 1}\\{11}\end{array}} \right]_{2 \times 1}}\]
Now, we will find \[\left( {B'A} \right)\left( {AB} \right)\].
 \[\left( {B'A} \right)\left( {AB} \right) = {\left[ {\begin{array}{*{20}{c}}{ - 7}&1\end{array}} \right]_{1 \times 2}} \times {\left[ {\begin{array}{*{20}{c}}{ - 1}\\{11}\end{array}} \right]_{2 \times 1}}\]
On multiplying these two matrices, we get
 \[ \Rightarrow \left( {B'A} \right)\left( {AB} \right) = {\left[ {\left( { - 7} \right) \times \left( { - 1} \right) + 1 \times 11} \right]_{1 \times 1}}\]
On further simplification, we get
 \[ \Rightarrow D = \left( {B'A} \right)\left( {AB} \right) = {\left[ {18} \right]_{1 \times 1}}\]
Therefore,
 \[ \Rightarrow n\left( D \right) = 1 \times 1 = 1\] …….. \[\left( 4 \right)\]
Now, we will find the determinant of matrix \[D\] i.e. \[\left| D \right|\] but we also know that the determinant of the matrix of order \[1 \times 1\] is equal to its element.
Therefore,
 \[\left| D \right| = 18\]
Now, we will calculate the square of it.
 \[ \Rightarrow {\left| D \right|^2} = {\left( {18} \right)^2}\]
On further simplification, we get
 \[ \Rightarrow {\left| D \right|^2} = 324\] …….. \[\left( 5 \right)\]
Now, we will calculate the value of the given expression i.e. \[\left[ {\dfrac{{n\left( C \right)\left( {{{\left| D \right|}^2} + n\left( D \right)} \right)}}{{n\left( A \right) - n\left( B \right)}}} \right]\]
Now, we will substitute all the values calculated in equation 1, equation 2, equation 3, equation 4 and equation 5.
 \[\left[ {\dfrac{{n\left( C \right)\left( {{{\left| D \right|}^2} + n\left( D \right)} \right)}}{{n\left( A \right) - n\left( B \right)}}} \right] = \dfrac{{4 \times \left( {324 + 1} \right)}}{{4 - 2}}\]
On simplifying the terms, we get
 \[ \Rightarrow \left[ {\dfrac{{n\left( C \right)\left( {{{\left| D \right|}^2} + n\left( D \right)} \right)}}{{n\left( A \right) - n\left( B \right)}}} \right] = 2 \times 325 = 650\]
Hence, the required value of the given expression is equal to 650.

Note: Matrix is defined as a set of numbers which are arranged in a particular row and columns. We need to keep in mind that when we multiply two matrices, then the matrix obtained by their product contains a number of rows equal to the number of rows of the first matrix and number of columns equal to the number of columns of the second matrix. Multiplication of matrices is not commutative.