Need help 9-12

aij(t) = mijt + bij. For example, here is a straight line of 2 x 2 matrices: B (t ) = For every value of t, we get a 2 x 2 matrix by plugging in t. For example, if you plug in t = 1 you get the matrix B(1) = 1 3 You can take the determinant of a straight line of matrices just like for an ordinary matrix, treating t as a variable: det B(t) = (1 - t)(4) - (3t) (t) = -3+2 - 4t + 4. Plug in t = 1 and you get det B(1) = -3 -4+1= -3, for example. (9) If A(t) is the straight line of matrices 2 2t 3 A (t ) = 6 0 1 - t 3 find det(A(t)) as a function of t. (10) If A(t) is any straight line of 2 x 2 matrices, explain why det A(t) is a polynomial function in t. (Hint: A(t) can be written as A(t) [a1 + a2t b1 + bat] [ci + cat di + d2t] for some real numbers a1, a2, b1, b2, C1, C2, di, and d2; then use the ad - bc formula we have for the determinant of a 2 x 2 matrix.) (11) If A(t) is any straight line of 3 x 3 matrices, explain why det A(t) is a polynomial function in t. (Hint: use cofactor expansion to express det A(t) in terms of determinants of straight lines of 2 x 2 matrices. By the previous problem, we know those determinants are polynomials, and you can use this to solve the 3 x 3 case.) (12) If det A(t) is a polynomial function in t, and there is some value of t such that A(t) is invertible, explain why A(t) is invertible for all but finitely many choices of t. (Hint: if f(t) is a degree-n polynomial, then f (t) has at most n roots.)