Q43. - Algebra 2

Question

Find the values of a, b, c, and d to make the statement $\left[\begin{array}{cc}3& 5\\ -1& 7\end{array}\right]\cdot \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -1& 7\end{array}\right]$ true. If the matrix$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is multiplied by any other matrix containing two columns, what do you think the result would be?

Step-by-Step Solution

Answer

The required value of a, b, c and b are 1, 0, 0 and 1 respectively. The multiplication will return the same matrix that is multiplied by the unknown matrix because it is an identity matrix.

Step 1- Simplify the left side of the provided equation

Perform multiplication between the matrices that are on left side of the provided equation. 

$\left[\begin{array}{cc}3\cdot a+5\cdot c& 3\cdot b+5\cdot d\\ (-1)\cdot a+7\cdot c& (-1)\cdot b+7\cdot d\end{array}\right]=\left[\begin{array}{cc}3a+5c& 3b+5d\\ -a+7c& -b+7d\end{array}\right]$

Step 2 – Extract the value of c and d

Equate the obtained matrix with the matrix on the right of provided equation.

$\left[\begin{array}{cc}3a+5c& 3b+5d\\ -a+7c& -b+7d\end{array}\right]=\left[\begin{array}{cc}3& 5\\ -1& 7\end{array}\right]$

Make two equations and extract equation for a and b from them.

$\begin{array}{c}3a+5c=3\\ 3a=-5c+3\\ a=\frac{-5c+3}{3}\end{array}$

$\begin{array}{c}3b+5d=5\\ 3b=-5d+5\\ b=\frac{-5d+5}{3}\end{array}$

Pick the equations$-a+7c=-1$ and$-b+7d=7$ after it substitute the values of a and b into these equations.

$\begin{array}{c}-\left(\frac{-5c+3}{3}\right)+7c=-1\\ \frac{5c-3}{3}+7c=-1\\ c=0\end{array}$

$\begin{array}{c}-\left(\frac{-5d+5}{3}\right)+7d=7\\ \frac{5d-5}{3}+7d=7\\ d=1\end{array}$

The obtained value of c and d are 0 and 1 respectively.

Step 3 – Extract the value of a and b

Substitute the obtained value of c and d into $a=\frac{-5c+3}{3}$and$b=\frac{-5d+5}{3}$respectively then simplify.

$\begin{array}{c}a=\frac{-5\cdot \left(0\right)+3}{3}\\ =\frac{3}{3}\\ =1\end{array}$

$\begin{array}{c}b=\frac{-5\left(1\right)+5}{3}\\ =0\end{array}$

The obtained value of a and b are 1 and 0 respectively.

Step 4- Make the conclusion

The matrix will be$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ . As it is an identity matrix so whenever a multiplication performed between this matrix and any other matrix having 2 columns then the result will be same as that matrix with two columns.