7. Matrix A is said to be involutory if A² = I. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.

Q: Find u and usuch that the null space for the matrix U= 3 v= 0 -2 1 1 -3 -15 1 - 9 1 is span{u, }.

A:

Q: Find an invertible matrix U such that UA R is the [21 1 -1 -1 0] -12 1 1 A = 3 -1 1-2 3

A:

Q: Show that the three points (x1, y1), (x2, y2), and (x3, y3) in a plane are collinear if and only if…

A:

Q: Show that eA is nonsingular for any diagonalizable matrix A.

A:

Q: Suppose A is a complex matrix. Show that AA and A" A are Hermitian.

A: Explanation of the answer is as follows

Q: If A is an orthogonal matrix and c is a scalar, then cA is orthogonal. True or False?

A: We have given a statement about orthogonal matrix and we have to state if it is true or false. For…

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: Prove that matrix Y is obtained from matrix X. Identify the property of determinants and prove |Y| =…

A: Y=4-984261-25X=1-3-74261-25 Y is obtained from X as shown below.

Q: 3. Let vi = [1, -2,3], v2 = [-1,–4,5], and v3 = [5,-1,3] be vectors in R³. a) Use determinant to…

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Q: Construct a 3 x 3 nonzero matrix A such that the vector is a solution of Ax = 0.

A: Given conditionAx = 0          .................(1)x = 111 is a solution to Ax=0,Let matrix A be - A…

Q: If a vector is in the range of a matrix A, then it is also in the null space of A. true of false?

A: Answer

Q: If the ows of a square matrix A are linearly dependent then A is singular

A: To determine whether the given statement is true or false.

Q: Explain why for any matrix A, AA' exists and prove that AA' is a symmetric matrix

A: Let A is any matrix of order m by n Am×n    and   Atn×m Now  AAt exists of order m by m To prove…

Q: 1) Suppose that A² = A and B is a diagonal matrix similar to A. Prove that B2 = B and that the…

A:

Q: Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are…

A: Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are…

Q: If the matrices A and B are conformable for sum A+B the product AB then prove that (A+B)" = A" +B"…

A: Prove that. (a).A+BT=AT+BT(b).ABT=BTAT

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.