Types Of Matrices

We are given that A, B, and C are singular matrices, meaning their determinants are equal to zero. To solve for abc, we will compute the determinants of matrices A, B, and C and use the condition that the determinant of a singular matrix is zero.

For matrix A, the determinant is:

det(A) = (a + b + c)(c) - (c + 1)(a + c).

Expanding this expression:

det(A) = ac + bc + c² - ac - c² - a - c = bc - a - c.

Since A is singular, we set the determinant equal to zero:

bc - a - c = 0 → bc = a + c.

For matrix B, the determinant is:

det(B) = (1)(2) - (3)(4) = 2 - 12 = -10.

Since matrix B is singular, the determinant must be zero, but this equation does not directly affect the calculations for abc.

For matrix C, the determinant is:

det(C) = (3b)(2) - (5)(a) = 6b - 5a.

Since matrix C is singular, the determinant is zero:

6b - 5a = 0 → .

Substitute into the equation from matrix A:

Multiply through by 6:

5ac = 6a + 6c.

Rearrange the equation:

5ac - 6a - 6c = 0.

Factor the equation:

a(5c - 6) = 6c.

Solve for a:

.

Now, multiply a, b, and c to find abc. Substituting into the equation for a, we find:

abc = 45.