Find the values of a, b, c, and d that make the matrix equation true.
-

Question

Quizplus · Accepted Answer

To find the values of a, b, c, and d, we need to multiply the matrices on both sides of the equation and equate the corresponding entries. 
Multiplying the left-hand side, we get:

|a - b|   |1 2|   |1|
|c + d| = |3 4| * |2| 
          |5 6|   |3|

Equating the corresponding entries with the right-hand side, we get the system of equations:

1a - 2c + 1b + 2d = 1
3a - 4c + 2b + 4d = -2
5a - 6c + 3b + 6d = 3

Solving this system of equations, we get a = 2, b = -1, c = -1, and d = 1.

Substituting these values back into the original equation, we can check that the equation is true.

The correct choice is therefore C: 11ebb17b_29fd_1e9e_afdc_1f51c1df743d_TB8591_11. To find the values of a, b, c, and d that make the matrix equation true, we need to solve for each entry of the resulting matrix.

The first row, first column entry is 2a + 3b. This must be equal to 17, so we have:

2a + 3b = 17

The first row, second column entry is 4a + 5b. This must be equal to 23, so we have:

4a + 5b = 23

The second row, first column entry is a + b + 2c + 3d. This must be equal to 11, so we have:

a + b + 2c + 3d = 11

The second row, second column entry is 2a + 3b + 4c + 5d. This must be equal to 19, so we have:

2a + 3b + 4c + 5d = 19

We can now use a system of equations to solve for a, b, c, and d. One way to do this is through substitution or elimination.

Solving for a and b, we get:

a = 4 - (3/2)b

Substitute this into the second equation:

4(4 - (3/2)b) + 5b = 23

Simplify and solve for b:

8 - 6b + 5b = 23

-b = 15

b = -15

Substitute b = -15 into the first equation to find a:

2a + 3(-15) = 17

2a = 62

a = 31

Now that we have a and b, we can solve for c and d using the remaining two equations.

Substitute a = 31 and b = -15 into the third equation:

31 - 15 + 2c + 3d = 11

2c + 3d = -5

Substitute a = 31 and b = -15 into the fourth equation:

2(31) + 3(-15) + 4c + 5d = 19

4c + 5d = -13

Using a system of equations with these two remaining equations, we can solve for c and d.

Multiply the third equation by 4 and subtract it from the fourth equation:

4c + 5d - 8c - 12d = -13 - (-20)

-4c - 7d = 7

Now we have two equations:

2c + 3d = -5

-4c - 7d = 7

Multiplying the first equation by 2 and adding it to the second equation:

-8c - 14d = 14

-7d = -12

d = 12/7

Substitute d = 12/7 into the first remaining equation to solve for c:

2c + 3(12/7) = -5

c = -41/14

So the values of a, b, c, and d that make the matrix equation true are:

a = 31, b = -15, c = -41/14, d = 12/7

Therefore, the correct answer is C.

Find the Values of A, B, C, and D That