Simplify the Matrix

Problem

[[−4,2*k+30],[−12,2*k−10]]*[[−4,24],[−k−15,2*k−10]]

Solution

1. Identify the matrix multiplication rule for a 2×2 matrix, where the element in row i and column j is the dot product of the ith row of the first matrix and the jth column of the second matrix.

2. Calculate the element at row 1, column 1 by multiplying the first row of the first matrix by the first column of the second matrix.

(−4)*(−4)+(2*k+30)*(−k−15)

16−2*k2−30*k−30*k−450

−2*k2−60*k−434

3. Calculate the element at row 1, column 2 by multiplying the first row of the first matrix by the second column of the second matrix.

(−4)*(24)+(2*k+30)*(2*k−10)

−96+4*k2−20*k+60*k−300

4*k2+40*k−396

4. Calculate the element at row 2, column 1 by multiplying the second row of the first matrix by the first column of the second matrix.

(−12)*(−4)+(2*k−10)*(−k−15)

48−2*k2−30*k+10*k+150

−2*k2−20*k+198

5. Calculate the element at row 2, column 2 by multiplying the second row of the first matrix by the second column of the second matrix.

(−12)*(24)+(2*k−10)*(2*k−10)

−288+4*k2−20*k−20*k+100

4*k2−40*k−188

6. Combine the calculated elements into the resulting matrix.

Final Answer

[[−4,2*k+30],[−12,2*k−10]]*[[−4,24],[−k−15,2*k−10]]=[[−2*k2−60*k−434,4*k2+40*k−396],[−2*k2−20*k+198,4*k2−40*k−188]]

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