S6

Question 1

If A and B are both n×n, then det(A+B) = det(A) + det(B).

Answer 1

False

Take for instance A and B.

Then det(A) + det(B) = 0 + 0 = 0, but det(A + B) = 1.

Question 2

If A and B are row equivalent, then they have the same determinant.

Answer 2

False

If B is obtained from A by a row interchange, or by multiplying a row, then they are row equivalent, but have different determinant.

Question 3

The linear transformation with matrix A is injective if and only if det(A) ≠ 0.

Answer 3

True

In class we saw that if det(A) ≠ 0, then A is invertible, and by the Invertible Matrix Theorem that we saw earlier, this is equivalent to the transformation defined by A being injective (if A is square).

Question 4

The determinant of the partitioned matrix is det(A) · det(D).

Answer 4

True

We can get this matrix into echelon form by separately putting A and D into echelon forms A_ech and D_ech.

Then this is an upper triangular matrix, so the determinant is the product of the diagonal entries. But we can split this product into the diagonal entries from A_ech and those from D_ech, which are exactly det(A) and det(D).

Question 5

The determinant of this partitioned matrix is det(A) · det(D)−det(B) · det(C).

Answer 5

False

Take for instance A = C = D = I₂. Then we can take basically anything for B and the equation will fail: