Answer to Question #101513 in Linear Algebra for Franco Yap

Question

C01

1.If Matrix A is 4 x 2, B is 3 x 4, C is 2 x 4, and D is 4 x 3, what is the size of the given expression.

A^TD + CB^T

2. Find the value of the k that makes the system 15 3 6 inconsistent.
-10 k 9

3.What is the second row of the product AB?

0 2 3 2 1 7
A = 5 4 8 B = 6 3 2
9 7 2 2 9 7

4. A is a 3 x 3 matrix and det(A)=7. what is det(2A).

Answer 1

1. A is 4x2, so A^T is 2x4(transposing). D is 4x3. A^T * D is multiplying 2x4 on 4x3, so its dimension would be 2x3. C is 2x4, B is 3x4, so B^T is 4x3. CB^T is multiplying 2x4 on 4x3, so its dimension would be 2x3. Sum of two 2x3 matrices is a 2x3 matice. Answer is 2x3.

2. System is called inconsistent when there is no solution, for example system x+y=1; x+y=2 is inconsistent because two parts of the systems contradict each other. In our case 15x+3y = -6, -10x+ky = -9. Multiply second by -1.5 and get 15x-1.5ky = 13.5.

-1.5k=3, k=-2.

Answer is -2.

3. Second row of product would be a result of multiplying second row of A on all columns of B:

2*6 + 1*3 + 7*2 = 29

2*9 + 1*7 + 7*2 = 39

2*2 + 1*9 + 7*7 = 62

Answer is 29 39 62.

4. det(cA) = c^n * det(A) where n is a dimension of A(nxn). In our case it's 3x3, so det(2A) = 2^3 * det(A) = 8 * det(A) = 8 * 7 = 56. Answer is 56