Three of four numbers have a sum of 22. If the average of the four numbers is 8, what is the fourth number ?

Compute the product AB by the definition of the product of matrices, where A and A are computed separately, and by the row-co

chubhunter [2.5K]

Complete Question

The complete question is shown on the first uploaded image

Answer:

First question

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}5\\-2\\\end{array}\right]$

Second question

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}5\\-2\\\end{array}\right] = \left[\begin{array}{ccc}{(-1 * 5 )+ (3* -2)}\\{(1 * 5)+ (4 * -2)}\\{(5 * 5) + (8*-2)}\end{array}\right] = \left[\begin{array}{ccc}{-11}\\{-3}\\{29}\end{array}\right]$

Third question

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}-3\\4\\\end{array}\right]$

Fourth question

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}-3\\4\\\end{array}\right] = \left[\begin{array}{ccc}{(-1 * -3 )+ (3* 4)}\\{(1 * -3)+ (4 * 4)}\\{(5 * -3) + (8*4)}\end{array}\right] = \left[\begin{array}{ccc}{15}\\{13}\\{-23}\end{array}\right]$

Fifth question

The correct option is A

Step-by-step explanation:

From the question we are told that

The matrix A is $A = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]$

The matrix B is $B = \left[\begin{array}{ccc}5&{-3}\\{-2}&4\end{array}\right]$

The first question is to set up the product $Ab_1$ , where $b_1$ is the first column of matrix B, this shown as

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}5\\-2\\\end{array}\right]$

The second question is to calculate $Ab_1$ , this is evaluated as

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}5\\-2\\\end{array}\right] = \left[\begin{array}{ccc}{(-1 * 5 )+ (3* -2)}\\{(1 * 5)+ (4 * -2)}\\{(5 * 5) + (8*-2)}\end{array}\right] = \left[\begin{array}{ccc}{-11}\\{-3}\\{29}\end{array}\right]$

The third question is to set up the product $Ab_2$ , where $b_2$ is the second column of matrix B, this shown as

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}-3\\4\\\end{array}\right]$

The fourth question is to calculate $Ab_2$ , this is evaluated as

$Ab_1 = \left[\begin{array}{ccc}{-1}&{3}\\ 1 &4 \\5 &8\end{array}\right]\left[\begin{array}{ccc}-3\\4\\\end{array}\right] = \left[\begin{array}{ccc}{(-1 * -3 )+ (3* 4)}\\{(1 * -3)+ (4 * 4)}\\{(5 * -3) + (8*4)}\end{array}\right] = \left[\begin{array}{ccc}{15}\\{13}\\{-23}\end{array}\right]$

The fifth question is to determine the numerical expression for the first entry in the first column of AB using the row-column rule and from the calculation of $Ab_1$ we see that it is

${(-1 * 5 )+ (3* -2)}$