a) Let A be a square matrix. If A = pB and B =qAt for some matrix B and numbers p and q, show that A=0= B or pq = 1. b) If A is a 2 x 2 matrix, show that A¹A = AA if and only if A is symmetric or b a A: [43] ]. -b a (Note that showing 'p if and only if q' means that you should show that p leads to q and q leads to p.)

Q: 6. Find the matrix of the quadratic form. Assume x is in R³. a. 3x} – 2x3 + 5.x + 4x1x2 – 6x1X3 b.…

A:

Q: Let A be a 4×5 matrix with 4 pivots. Do the columns of A span R4? Do they span R5? Justify your…

A: As A is a 4 x 5 matrix with 4 pivots so, it has a pivot in its every row. So, yes the columns of A…

Q: Exercise 51. Is there a square matrix X with real entries such that det(X¹X) = -3?

A: detXTX=-3 X is a square matrix with real entries.

Q: b) Consider the following matrices: Solution: M₁ = [1 −7], M₂ = [1 2], M₁ = (1 1]. A = [23], B= [13…

A: Step 1: Determine if A is in the set Span({M1​,M2​,M3​}).Step 2: Determine if B is in the set…

Q: diag(a1, ..., an) be a diagonal matrix (i.e., the matrix with the numbers a1, ..., an on the main…

A:

Q: Let A E Rx and B E Rmxm and assume that dim N(A) = k and dim N(B) = l. Let C be an 2 x 2 block…

A:

Q: 24. Let A [A₁, A2, A3] be a (3 × 3) matrix and set = 2 a) Show that B - 3 1 00 -1 ] 3 4 AB = [2A₁ +…

A: Here, the given A=A1, A2, A3 be a 3×3 matrix and set  B=2003-10134.