Several operations such as addition, subtraction, multiplication, etc. can be performed on matrices. The value of matrix B is:
[tex]B =\left[\begin{array}{c}1&3&-5\end{array}\right][/tex]
Given that:
[tex]A =\left[\begin{array}{ccc}2&4&-2\\4&-5&7\\2&7&5\end{array}\right][/tex]
[tex]AB =\left[\begin{array}{c}24&-46&-2\end{array}\right][/tex]
Let matrix B be:
[tex]B =\left[\begin{array}{c}a&b&c\end{array}\right][/tex]
The product of A and B is as follows:
[tex]\left[\begin{array}{ccc}2&4&-2\\4&-5&7\\2&7&5\end{array}\right] \times\left[\begin{array}{c}a&b&c\end{array}\right][/tex]
Multiply the rows of A to the column of B. So, we have:
[tex]AB =\left[\begin{array}{ccc}2a+4b-2c\\4a-5b+7c\\2a+7b+5c\end{array}\right][/tex]
This means that:
[tex]\left[\begin{array}{ccc}2a+4b-2c\\4a-5b+7c\\2a+7b+5c\end{array}\right] = \left[\begin{array}{c}24&-46&-2\end{array}\right][/tex]
By comparison, we have:
[tex]2a + 4b - 2c = 24[/tex] --- (1)
[tex]4a - 5b + 7c = -46[/tex] --- (2)
[tex]2a + 7b + 5c = -2[/tex] --- (3)
Multiply (1) and (3) by 2
[tex]2a + 4b - 2c = 24[/tex] --- (1)
[tex]4a + 8b - 4c = 48[/tex] ----- (5)
[tex]2a + 7b + 5c = -2[/tex] --- (3)
[tex]4a + 14b + 10c = -4[/tex] ---- (6)
So, we have:
[tex]4a - 5b + 7c = -46[/tex] --- (2)
[tex]4a + 8b - 4c = 48[/tex] ----- (5)
[tex]4a + 14b + 10c = -4[/tex] ---- (6)
Make 4a the subject in each of the equations
[tex]4a = -46 + 5b - 7c[/tex]
[tex]4a = 48 - 8b + 4c[/tex]
[tex]4a = -4 - 14b - 10c[/tex]
[tex]4a = 4a[/tex]. So, we have:
[tex]-46 + 5b - 7c = 48 - 8b + 4c[/tex] and [tex]-46 + 5b - 7c = -4 - 14b - 10c[/tex]
Simplify both expressions
[tex]-46 + 5b - 7c = 48 - 8b + 4c[/tex]
[tex]5b + 8b - 7c - 4c = 48 + 46[/tex]
[tex]13b - 11c = 94[/tex]
[tex]-46 + 5b - 7c = -4 - 14b - 10c[/tex]
[tex]5b+14b - 7c + 10c = 46-4[/tex]
[tex]19b + 3c = 42[/tex]
Make c the subject
[tex]c = \frac{42 -19b}{3}[/tex]
Substitute [tex]c = \frac{42 -19b}{3}[/tex] in [tex]13b - 11c = 94[/tex]
[tex]13b - 11 \times (\frac{42 - 19b}{3}) = 94[/tex]
Multiply through by 3
[tex]39b - 11 \times (42 - 19b) = 282[/tex]
Open bracket
[tex]39b - 462 + 209b = 282[/tex]
Collect like terms
[tex]39b + 209b = 282+462[/tex]
[tex]248b = 744[/tex]
Solve for b
[tex]b = \frac{744}{248}[/tex]
[tex]b = 3[/tex]
Solve for c
[tex]c = \frac{42 -19b}{3}[/tex]
[tex]c = \frac{42 - 19 \times 3}{3}[/tex]
[tex]c = \frac{-15}{3}[/tex]
[tex]c = -5[/tex]
Solve for a in [tex]4a = -4 - 14b - 10c[/tex]
[tex]4a = -4 - 14 \times 3 - 10 \times -5[/tex]
[tex]4a = 4[/tex]
Divide by 4
[tex]a = 1[/tex]
Recall that matrix B is:
[tex]B =\left[\begin{array}{c}a&b&c\end{array}\right][/tex]
Hence, the value of matrix B is:
[tex]B =\left[\begin{array}{c}1&3&-5\end{array}\right][/tex]
Read more on matrix operations at:
https://brainly.com/question/16956653
