1.4 34 and 46 please on paper thanks a lot 41.RC = tr(CR).42. If A is a square matrix and n is a positive integer, is it true that(A)T = (AT)"? Justify your answer.43. a. Show that if A is invertible and AB = AC, then B = C.b. Explain why part (a) and Example 3 do not contradict oneanother.44. Show that if A is invertible and k is any nonzero scalar, then(KA)" = k"A" for all integer values of n.45. a. Show that if A, B, and A + B are invertible matrices withthe same size, thenA(A¯¹ + B¯¹)B(A + B)-¹ = Ib. What does the result in part (a) tell you about the matrixA¹+B-¹?46. A square matrix A is said to be idempotent if A² = A.a. Show that if A is idempotent, then so is I - A.b. Show that if A is idempotent, then 2A - I is invertible andis its own inverse.47. Show that if A is a square matrix such that Ak = 0 for somepositive integer k, then the matrix I-A is invertible and(I - A)¹ = I + A + A²++ Ak-148. Show that the matrix^=[ 2Asatisfies the equationA² - (a + d)A + (ad - bc)I = 049, Assuming that all matrices are nxn and invertible, solve 29. The matrix A in Exercise 21.30. An arbitrary square matrix A.31. a. Give an example of two 2 x 2 matrices such that(A + B)(A - B) # A² - B²b. State a valid formula for multiplying out(A + B)(A - B)c. What condition can you impose on A and B that will allowyou to write (A + B)(A - B) = A² - B²?32. The numerical equation a² : = 1 has exactly two solutions. Findat least eight solutions of the matrix equation A² = 13. [Hint:Look for solutions in which all entries off the main diagonalare zero.]433. a. Show that if a square matrix A satisfies the equationA² + 2A + I = 0, then A must be invertible. What is theinverse?b. Show that if p(x) is a polynomial with a nonzero constantterm, and if A is a square matrix for which p(A) = 0, thenA is invertible.34. Is it possible for A³ to be an identity matrix without A beinginvertible? Explain.35. Can a matrix with a row of zeros or a column of zeros have aninverse? Explain.a.b47.48.

Question

1.4   34 and 46 please on paper thanks a lot 41.RC = tr(CR).42. If A is a square matrix and n is a positive integer, is it true that(A)T = (AT)&#34;? Justify your answer.43. a. Show that if A is invertible and AB = AC, then B = C.b. Explain why part (a) and Example 3 do not contradict oneanother.44. Show that if A is invertible and k is any nonzero scalar, then(KA)&#34; = k&#34;A&#34; for all integer values of n.45. a. Show that if A, B, and A + B are invertible matrices withthe same size, thenA(A¯¹ + B¯¹)B(A + B)-¹ = Ib. What does the result in part (a) tell you about the matrixA¹+B-¹?46. A square matrix A is said to be idempotent if A² = A.a. Show that if A is idempotent, then so is I - A.b. Show that if A is idempotent, then 2A - I is invertible andis its own inverse.47. Show that if A is a square matrix such that Ak = 0 for somepositive integer k, then the matrix I-A is invertible and(I - A)¹ = I + A + A²++ Ak-148. Show that the matrix^=[ 2Asatisfies the equationA² - (a + d)A + (ad - bc)I = 049, Assuming that all matrices are nxn and invertible, solve 29. The matrix A in Exercise 21.30. An arbitrary square matrix A.31. a. Give an example of two 2 x 2 matrices such that(A + B)(A - B) # A² - B²b. State a valid formula for multiplying out(A + B)(A - B)c. What condition can you impose on A and B that will allowyou to write (A + B)(A - B) = A² - B²?32. The numerical equation a² : = 1 has exactly two solutions. Findat least eight solutions of the matrix equation A² = 13. [Hint:Look for solutions in which all entries off the main diagonalare zero.]433. a. Show that if a square matrix A satisfies the equationA² + 2A + I = 0, then A must be invertible. What is theinverse?b. Show that if p(x) is a polynomial with a nonzero constantterm, and if A is a square matrix for which p(A) = 0, thenA is invertible.34. Is it possible for A³ to be an identity matrix without A beinginvertible? Explain.35. Can a matrix with a row of zeros or a column of zeros have aninverse? Explain.a.b47.48.

Accepted Answer

Answered: Image /qna-images/answer/55e215a7-02a2-47d1-98e9-85676d0fd7b8.jpg

A is invertible. Is it possible for A³ to be an identity matrix without A being invertible? Explain. Can a matrix with a row of zeros or a column of zeros have an inverse? Explain.