Let A = [a b c, d e f, g h i]. B = [9 3h i, 2d 6e 2f, a 3b c]. Suppose that det (A) = 2. Find (det (AB)^T). Answer is 24. How do I find the answer with the right steps?
If all the elements of a row or column of A are multiplied by k, then |A| → k|A|
Since The middle row of B is that of A multiplied by 2 and the middle column of B is multiplied by 3.
So, |B| = 2*3*|A|
That means |AB| = |A|*|B| = 2*12
and then |AB|T = -|AB|
Answer is -24.
Thank you!
To find the answer, we need to follow these steps:
Step 1: Find the determinant of matrix B, det(B).
Step 2: Calculate the product of the determinants of A and B, det(A) * det(B).
Step 3: Take the transpose of the product obtained in step 2, (det(A) * det(B))^T.
Step 4: Simplify the expression to get the final answer.
Let's go through the steps one by one:
Step 1: Find the determinant of matrix B, det(B).
The matrix B is given as:
B = [9 3h i, 2d 6e 2f, a 3b c]
Using the determinant expansion along the first row, we have:
det(B) = 9 * det([6e 2f, 3b c]) - 3h * det([2d 2f, a c]) + i * det([2d 6e, a 3b])
Expanding each of the submatrices, we have:
det(B) = 9 * (6e * c - 2f * 3b) - 3h * (2d * c - 2f * a) + i * (2d * 3b - 6e * a)
Simplifying further, we get:
det(B) = 54ec - 18fb - 6ahc + 6fha + 6idb - 18iae
Step 2: Calculate the product of the determinants of A and B, det(A) * det(B).
The determinant of matrix A is given as det(A) = 2.
Multiplying the determinants, we have:
det(A) * det(B) = 2 * (54ec - 18fb - 6ahc + 6fha + 6idb - 18iae)
Step 3: Take the transpose of the product obtained in step 2, (det(A) * det(B))^T.
Taking the transpose of the expression, we have:
(det(A) * det(B))^T = (2 * (54ec - 18fb - 6ahc + 6fha + 6idb - 18iae))^T
Since the expression is a scalar, the transpose does not affect it. Therefore, we can rewrite this as:
(det(A) * det(B))^T = 2 * (54ec - 18fb - 6ahc + 6fha + 6idb - 18iae)
Step 4: Simplify the expression to get the final answer.
Finally, we can simplify the expression:
(det(A) * det(B))^T = 108ec - 36fb - 12ahc + 12fha + 12idb - 36iae
The given answer is 24, so we set the expression equal to 24 and solve for the variables:
108ec - 36fb - 12ahc + 12fha + 12idb - 36iae = 24
Simplifying further, we get:
54ec - 18fb - 6ahc + 6fha + 6idb - 18iae = 12
Comparing this equation with the expression we obtained, we can see that the given answer of 24 is incorrect. The correct answer should be 12.