Question

For two $3\times 3$ matrices A and B , let A + B = 2B’ and 3A + 2B = $I_3$ where B’ is the transpose of B and $I_3$is $3\times 3$ identity matrix . Then
A) $5A+10B=2I_3$
B) $3A+6B=2I_3$
C) $10A+5B=3I_3$
D) $B+2A=I_3$

Answer 1

Hint:
Taking transposes on the given equations we get $3A^{{}^{\prime }}+2B^{{}^{\prime }}=I_3$ and $A^{{}^{\prime }}=2B-B^{{}^{\prime }}$ and simplifying further we get that A = B and $I_3$= 5A and using these values we can get the equation which satisfies these equations.

Complete step by step solution:
We are given that A + B = 2B’
Applying transpose on both sides
$\Rightarrow {\left(A+B\right)}^{{}^{\prime }}={\left(2B^{{}^{\prime }}\right)}^{{}^{\prime }}$
We know that ${\left(A+B\right)}^{{}^{\prime }}=A^{{}^{\prime }}+B^{{}^{\prime }}$ and ${\left(A^{{}^{\prime }}\right)}^{{}^{\prime }}=A$
From this we get ,
 $$\begin{gathered} \Rightarrow A^{{}^{\prime }}+B^{{}^{\prime }}=2B \\ \Rightarrow A^{{}^{\prime }}=2B-B^{{}^{\prime }} \end{gathered}$$
We are given that 3A + 2B = $I_3$
Applying transpose on both sides we get
$$\begin{gathered} \Rightarrow {\left(3A+2B\right)}^{{}^{\prime }}={\left(I_3\right)}^{{}^{\prime }} \\ \Rightarrow 3A^{{}^{\prime }}+2B^{{}^{\prime }}=I_3 \end{gathered}$$
Substitute the value of $A^{{}^{\prime }}$in the above equation
$$\begin{gathered} \Rightarrow 3(2B-B^{{}^{\prime }})+2B^{{}^{\prime }}=I_3 \\ \Rightarrow 6B-3B^{{}^{\prime }}+2B^{{}^{\prime }}=I_3 \\ \Rightarrow 6B-B^{{}^{\prime }}=I_3 \end{gathered}$$
We are given that A + B = 2B’
From this $B^{{}^{\prime }}={\displaystyle \frac{A+B}{2}}$
Substitute this in the previous equation
$$\begin{gathered} \Rightarrow 6B-\left({\displaystyle \frac{A+B}{2}}\right)=I_3 \\ \Rightarrow 12B-A-B=2I_3 \end{gathered}$$
Substitute the value of $I_3$from the given equation 3A + 2B = $I_3$
$$\begin{gathered} \Rightarrow 12B-A-B=2(3A+2B) \\ \Rightarrow 11B-A=6A+4B \\ \Rightarrow 7A=7B \\ \Rightarrow A=B \end{gathered}$$
Substituting in 3A + 2B = $I_3$we get
$$\begin{gathered} \Rightarrow 3A+2A=I_3 \\ \Rightarrow 5A=I_3 \end{gathered}$$
Now we have A = B and $I_3$= 5A
We have asked 5A +10B
$\Rightarrow 5A+10B=5A+10A=15A=3\times 5A=3I_3$

Therefore the correct option is C.

Note:
1) In mathematics, a matrix (plural: matrices) is a rectangular table of cells of numbers, with rows and columns.
2) Every square dimension set of a matrix has a special counterpart called an "identity matrix". The identity matrix has nothing but zeroes except on the main diagonal, where there are all ones.
3) An inverse matrix is a matrix that, when multiplied by another matrix, equals the identity matrix.