11. Complete each of the following sentences with the word "must", "might" or "cannot".If T : R → R' is a linear transformation, then T(a)be one-to-one.(b)AT Abe a symmetric matrix.(c)If B = B, then Bbe invertible.%3D(d)of C.For a square matrix C, the dimension of Nul(C).equal the determinant

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Answered: 11. Complete each of the following…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter6: Matrices And Determinants

Section: Chapter Questions

Problem 11CC

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The gray square in Table 1 has the following vertices: [00],[10],[11],[01] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c=2 in the expansion matrix and c=1 in the shear matrix.
Let T be a linear transformation from M2,2 into M2,2 such that T([1000])=[1102], T([0100])=[0211], T([0010])=[1201],T([0001])=[3110]. Find T([1314]).
Consider that T is R^3 -> R^3 is a linear transformation Is T is invertible and if so what is the formula for T^-1 if T is invertible?
If the determinant of a 5×5 matrix A is det(A)=8, and the matrix C is obtained from A by swapping the third and fourth rows, thendet(?)=
Why does every linear transformation T from R 2 to R 2 take squares to parallelograms? Rectangles also go to parallelograms (squashed if T is not invertible).
Label the following statements as true or false.(a) The function det: M2x2(F) -> F is a linear transformation.(b) The determinant of a 2 x 2 matrix is a linear function of each rowof the matrbc when the other row is held fixed.(c) If A E M2x2 (F) and det(A) = 0, then A is invertible.
17.Consider the linear system dxdt=a11x+a12y,dydt=a21x+a22y,dxdt=a11x+a12y,dydt=a21x+a22y, where a11, a12, a21, and a22 are real-valued constants. Let p = a11 + a22, q = a11a22 − a12a21, and Δ = p2 − 4q. Observe that p and q are the trace and determinant, respectively, of the coefficient matrix of the given system. Show that the critical point (0, 0) is a a.Node if q > 0 and Δ ≥ 0; b.Saddle point if q < 0; c.Spiral point if p ≠ 0 and Δ < 0; d.Center if p = 0 and q > 0. Hint: These conclusions can be reached by studying the eigenvalues r1 and r2. It may also be helpful to establish, and then to use, the relations r1r2 = q and r1 + r2 = p.
Label the following statements as true or false. (a) The function det: Mn×n(F) →F is a linear transformation. (b) The determinant of a square matrix can be evaluated by cofactor expansion along any row. (c) If two rows of a square matrix A are identical, then det(A) = 0. (d) If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = −det(A). (e) If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A). (f) If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = k det(A). (g) If A ∈ Mn×n(F) has rank n, then det(A) = 0. (h) The determinant of an upper triangular matrix equals the product of its diagonal entries.
If A is a 3rd order matrix with a determinant of 5, then det((2A)^-1)=?
How is the matrix of R^−1 obtained from that of R in general, that is, when A = {a1, . . . , am}, B = {b1, . . . , bn} and R is an arbitrary relation from A to B?
why no matrix exists for one-to-one linear transformation from R^4 to R^3
If the determinant of a 3×3 matrix A is det(A)=2, and the matrix B is obtained from A by multiplying the first column by 3, thendet(B)=

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