8. A square matrix P has a property that P² = P. If P# In, which of the following is true? (a) P is singular. (b) P is symmetric. (c) P is skew-symmetric. (d) P is an elementary matrix.

Q: 2. Let A be a square matrix with real or complex entries. Show that if A is normal and upper…

A: Definition of normal matrix:

Q: 1. Let, A = -1 21 Find the matrix -5A3 + 2A? + 4A + 512. 9 21

A: Given: A=-1292 Determine A3. A3=-1292-1292-1292=192922-1292=-14218962

Q: 5. For the following matrix A, decide whether the columns of A span R³. Show your work. A= 3-2 1 -2…

A:

Q: 1 Find a matrix P that satisfies P 1 9 1 2 0 3 0 900 23 1 7 1 1

A: We have to find the matrix P that satisfies the given equation.

Q: 5. Find the three by three matrix A such that [r – y] A. y for all r, y, z in R.

A: In the given problem we have to find the three by three matrix A such that…

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

A:

Q: Assume A is nonsingular. Prove that det(A-1) 1 det(A) Let c be any scalar. Prove that det(cA) = c"…

A:

Q: 19. Can each vector in R4 be written as a linear combination of the columns of the matrix A above?…

A:

Q: 5. Let D be a non-singular matrix such that 7. (7D)- 1 -2 Find D.

A: Inverse matrices

Q: Let A, B; be Square matrixes, which meet the following conditionS BA = A(A+ I) & A3 I %3D 1. Prove…

A: We will use properties of matrix multiplication

Q: AE4.6 Show that the pair (A, C) is observable if and only if the pair (A-LC, HC) is observable for…

A: Given that,   The pair A,C   n×p matrix L and any non-singular p×p matrix H.  Then , A-LC,HC is…

Q: 7. Let A and B two 3-by-3 matrices, with |B| = 1. Let C be the inverse matrix of B. Which of the…

A:

Q: 1 -4-2) s) Let A := -2 3 2 4 -8-5 (a) Show that A is diagonalizable and find a nonsingular P such…

A:

Q: 7. Let M be a 3 × 3 matrix. Prove that if M is upper triangular or lower triangular, then det (M) is…

A:

Q: matrix A if (A" – 21)* = } Determine the matrix A if %3D

A: Given,             12AT-2I-1=1234To find : The matrix A.

Q: 31. Let A be a 5 ×7 matrix with rank 4. a. What is the dimension of the solution space of Ax = 0? b.…

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Q: Find A if [: :]) 1 1 A 2 2 2

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Q: Q2) Find x for the following matrix 3 1 x| a) -1 3 4= -30 1 ol 1 |1 2 1| b) 12 x 2=0 13 6 xl

A: We need to solve the determinant and find the value of x.  Now, if we have a 3×3 matrix…

Q: Q5. Suppose B = QRQ-1, where Q is orthogonal, and R is upper triangular. Show that if B is…

A:

Q: Suppose that Q is a matrix with the property Q−1 =QT . Show thatqij = Qij/det(Q)

A: Given: Q-1 = QT To Prove: qij = Qij/det(Q) Formula Used: If A is a square matrix, then its inverse…

Q: (a) Describe the kernel and the image of Jn. (b) Use this to show that J, is diagonalisable when n #…

A:

Q: 7. Let A be a 32-by-48 real matrix. Which of the following matrices is always symmetric? (a) A+ A…

A: As per the company rule, we are supposed to solve one problem from a set of multiple problems.…

Q: 7.2. Let A be a matrix with the property that columns 1,3,5,7,... are orthogonal to columns 2, 4, 6,…

A:

Q: 8. Let A be an m x n matrix. Prove that if n is an odd number, then rank(A) = nullity(A).

A: Rank nullity Theorem : Nullity (A) + Rank(A)=Dim (Domain) =n

Q: 19. Let 2 A=012 [132] If possible, find a matrix B such that AB = A² + 2A.

A:

Q: the acdjainbaf square of As of 2.

A:

Q: 10. Consider the matrix A generated by the following expression A = eye(4) + ones(4) Which of the…

A: Here I have the code you can see option D is correct.

Q: 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: The answers are given below.

Q: Let p and q be nonnegative integers and let A be a square matrix. Show that Ap Aq = Ap+q and (Ap ) q…

A: We have, p and q be non-negative integers and let A be a square matrix. Consider ApAq. Since p and q…