a. Prove that if A and B are n xn non-singular matrices, then (AB)-1 = B-'A-1.

Q: Let B and C be n×n matrices with the property that Bx = Cx for all x ∈ Rn. Show that B = C.

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Q: t A and B be arbitary n x n matrices. If B is invertible, prove that AB- = B-'A if and only if AB =…

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Q: Let E and F be n × n elementary matrices and let C = EF. Is C nonsingular? Explain.

A: Let E and F are elementary matrices and let

Q: .

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Q: A and B are n X n matrices If Bis invertible, prove that det(B-1AB) = det(A)

A: A and B are n X n matrices and B is invertible.

Q: V2 1 2 11 T = 0- 2 000 -Vī

A: If A is a matrix of order and I is the identity matrix of order, then If A-1 is the inverse matrix…

Q: Suppose that A, B are n x n matrices. Prove that if A is singular, then B is singular.

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Q: Let A and B be n × n matrices such that AB = I. Prove that ∣A∣ ≠ 0 and ∣B∣ ≠ 0.

A: Since AB = I, therefore determinant of AB is 1, that is |AB|=1. This implies that AB is invertible.…

Q: 2) Suppose that A and B are nxn matrices that satisfy AB=AB. Show that AB = AB.

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Q: Prove that if A and B are similar matrices, then ∣A∣ = ∣B∣. Is the converse true?

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Q: 3. Find x such that the matrix A is equal to its own inverse. A = G %3D

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Q: Suppose A is a 2 x 2 matrix such that A and 1 A What is A-1?

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Q: Find elementary matrices e and e2 such that di 0 e2 = 1 5 1 5 e1 0 dz) for some numbers d and d2.

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Q: Consider A and B are two square matrices and B is a scalar. Calculate = ((BA^T B)^(-1 %3D \beta…

A: Matrix Properties: (c A)-1 = c-1 A-1 (AT)-1 = (A-1)T (AB)-1 = B-1A-1   Therefore; (βATB)-1 = β-1 B-1…

Q: Let A, D, and P be n × n matrices satisfying AP = PD. Assume that P is nonsingular and solve this…

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Q: Which of the following matrices is a nilpotent matrix of degree 2. a a) where ab =-x². -x a where ab…

A: We have to select the nilpotent matrices of order 2 from the above list.

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. If B is a singular matrix then…

A: We have given  det(A) = 3 If B is singular matrix  det(B)  = 0 detBT=det(B) =0 So,  det2A-2BT =0…

Q: 8. Prove that (i) ||7. =1, and (ii) |. =\n (I is the Identity Matrix of order n).

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Q: Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular.

A: Let A and B be n × n matrices such that AB is singular.

Q: 3. Suppose A E Mnxn(C) is a normal, invertible matrix. Prove that A* A-1 is unitary.

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Q: Let A and B be n × n matrices. If AB = A then B is the identity matrix In. True or False

A: To write that true or false

Q: a. Prove that if A and B are n x n non-singular matrices, then (AB)-1 = B-'A-1.

A: Note: As per our guidelines we are supposed to answer only 1️⃣ question. Kindly repost other…

Q: let A and B be n ×n invertible matrices. Prove that AB invertible and (AB)−1 = B−1A−1.

A: Let A and B be n×n invertible matrices, with inverses A–1 and B–1, respectively. Consider their…

Q: 4. Prove that the statement (XY)- = Y-1X-1 is true for any two square matrices, where their inverses…

A: We have to prove that the statement XY-1=Y-1X-1  is true for any two square matrices. Where their…

Q: 2. Assuming that all matrices are nxn and invertible, solve for D. CTB-A'BACDA²²B™C-2 = CT

A: We use properties of invertible matrices.

Q: Let A and B be two matrices of size 3 x 3 such that det(A) = -2 and det(B) = 3. Then det(2A-'B") =…

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Q: 4. Suppose B, I – B-' are invertible matrices and (X – A)- = BX-1, find X (as an expression in A,…

A: A matrix Y is said to be invertible if and only if Y-1 Y=Y Y-1=I

Q: 2. Verify, without using the matrix for 0(), that [O()]° is the identity matrix 1 0 0 1

A: This is a problem of rotation of the matrix.

Q: Let A and C be n×n matrices such that ​CA=I​ (the n×n identity​ matrix). Show that Ax​=0 has only…

A: Here, CA=I, where I is the identity matrix.

Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = * \beta…

A: Topic:- matrix operation  Note:- if more than one question uploaded than only first question need to…

Q: Find nonzero 2 × 2 matrices A and B such that AB = O.

A: Consider

Q: Find conditions on w, x, y, and z such that AB = BA for the matrices below.

A: Here we find the condition w,x,y,z such that AB=BA where A=wxyz , B=11-11

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. If B is a singular matrix then…

A: Let A and B be two matrices of size 4 x 4

Q: 1 4 What is a, if A = is a singular matrix ? %3D 2 a

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Q: Find two non zero matrices A and B such that AB = 0. %3|

A: Find the two matrix which satisfy the given condition

Q: Let A and B be two matrices of size 4 x 4 such that det(A) = 3. %3D If B is a singular matrix then…

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Q: Let A and B benxn matrices such that AB = 1. Prove that |A| +0 and |B| + 0.

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Q: Let A and B be n × n matrices and let C = AB. Prove that if B is singular then C must be singular.

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Q: 3. If A is a nonsingular matrix, then A' is nonsingular and its inverse is (A')1 = (A') '. Prove it!

A: A matrix A is said to be invertible if and only if  det(A) is not equal to zero i.e.  |A| ≠ 0

Q: 3. Prove that (i) /, =1, and (ii) ||, =Vn (I is the Identity Matrix of order n).

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Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = * \beta…

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Q: If A and B are singular n × n matrices, then A + Bis also singular.

A: If A and B are singular  matrices the A+B is also singular. (False Statements)

Q: If A is a positive definite matrix and B is positive semi-definite, show that trace (AB) 2 O with…

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Q: 16. Prove that for any square matrix A, ||A||ž E ||A* A||2.

A: Given: A is a square matrix.

Q: 6. Show that (AA)* = X* A* for all A E C and all matrices A.

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Q: Please see image

A: Given matrices are

Q: Let A, B, C be n × n matrices and assume that A is invertible. Solve the equation A(B + X) = C for…

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Q: Assume A, B, C are invertible matrices of appropriate dimension. Solve for X: AX+B=CX 1 OX= B(C-A)-¹…

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Q: Suppose A and I are bothn x n matrices, then

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Q: Show that there are no 2 X 2 matrices A and B such that AB - BA = I2

A: Given- There are 2×2 matrices A and B. To disprove- That AB-BA=I2

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.