1. Let A and B are two nxn matrices, and I is a nxn identity matrix. If I + AB has an inverse, prove: (I + BA) = I–B (I + AB)'A.

Q: [2 8) Use the Gauss-Jordan method to find the inverse of the matrix A: %3D

A:

Q: Show that if A and B are positive definite matrices, then A® B is positive definite.

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Q: Use Theorem 1.4.5 to compute the inverse of the following matrix. A-8 | NOTE: Write the elements of…

A:

Q: Prove that if both products AB and BA are defined, then AB and BA are square matrices. Assume that A…

A: We want to prove that AB and BA are square matrices

Q: Prove Any n by n triangular matrix with main diagonal entriesare all 1 is diagonalizable if and only…

A: Triangular Matrix: A triangular matrix is a square matrix where all the values above or below the…

Q: Let A and B be n × n nonsingular matrices. Find nonsin- gular matrices P and Q so that PAQ = B.

A: Given that A and B  are n×n non singular matrices. An n×n matrix A is non singular only if A is…

Q: Find a 2 x 1 matrix x with entries not all zero such that 2 Ax = 3x, where A = 1 2

A: Given matrix is A=2   11   2 and condition is Ax=3x

Q: Suppose A. B and X are 'nxn matrices such that A and B are invertible, and Which one of the…

A:

Q: 6.* Let C be a nonsingular matrix with 3 rows and 2 columns. Explain why C has a left inverse.

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Q: If A and C arę nxn invertible matrices and B is an nxn matrix, then

A: By Theorem we can say that

Q: Construct 3 × 2 matrices A and B such that Ax = 0 has only the trivial solution and Bx = 0 has a…

A:

Q: 6. Show that if Q is an orthogonal matrix, then 1 0 %3D 0 Q is also an orthogonal matrix.

A: We have to show that HHt = I

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

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Q: Show that these four matrices form a representation of the Vierergruppe, whose multi- plication…

A: 17.2.2 We have to show that given matrices  I,A,B and C form a Vierergruppe. Concept:…

Q: Express the following matrix as a product of several elemen- tary matrices. 1 3 E 2 3 ||

A:

Q: 5. Verify that the product of the non-singular matrix and its inverse from the previous problem is…

A: (5) Given matrix A=32−22 and its inverse is A−1=1102−223. We have to verify that AA−1=I2…

Q: Find inverse of matrix A by Gauss-Jordan me [2 -1 2

A: Here we have to find the inverse of the matrix by the Gauss-Jordan method.  we have,

Q: If A and B are square matrices of the same size such that AB = BA, prove that (AB)2 = A2B2.  By…

A:

Q: 2) If IAI = 3, where A is a 3x3 matrix, caleulate the folowing: 2a) det ( ZA) 26) det (A) 2c) det (…

A:

Q: Suppose A and B are orthogonal matrices. Show that A", A-1, AB are also orthogonal.

A: we have given A and B are orthogonal matrices.  We have to show that the following

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: 2 21 Compute the adjoint matrix and inverse matrix of the following matrix: -1 1 2 1 1 If the…

A: Explanation of the answer is as follows

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 be an nxn matrix and B = A+1, where I is the identity matrix, then it is possible for A and B…

A: Given A be an n×n matrix nd B=A+I where I is the identity matrix , then it is possible for A and B…

Q: Let A be annxn matrix and B=A+T, where / is the identity matrix, then it is possible for A and B to…

A: Given matrix combination is B = A + I

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. Determine the matrix X, such that:

A: Solution:-

Q: 4. If A is a nonsingular nxn matrix, show that AA¹ is nonsingular and det(AA) > 0.

A: Given A is a nonsingular matrix.

Q: If A and B are orthogonal nxn matrices, then AB-1 is an orthogonal matrix. True False

A: here we must know that  1)  AA-1=A-1A =I2) ATT=A3) AI =IA =A

Q: If A and B are 3×3 invertible matrices, such that det(A)=2, det(B) =-2. Then det (ABA^T) = ???

A: To solve this problem, we use the standard result of matrix

Q: 1. For any 3x3 matrices A & B: Verify that (AB)" = B"A". %3D

A: By using the properties

Q: 4 2 If A is a nonsingular matrix whose inverse is find A.

A: If A is a nonsingular matrix whose inverse is 4211. we have to find A.   the inverse of the matrix A…

Q: 4. Prove: If A and B are both n x n matrices, then det(AB)= detAdetB. (Only use 2x2 matrices)

A: Given A and B  n×n  matrices to prove  det(AB)=det A det B

Q: 3- If A² [45] [95] 091 (1 find A where A is UPF triangula matrix

A:

Q: Let A be annxn matrix and B= A+T, where I is the identity matrix, then it is possible for A and B to…

A: Let λ be the eigen value of A. As B=A+I So, eigen value of B will be λ+1 which is not same as λ

Q: 6 X1 -1 -5 X2 II

A: Let us consider the system of differential equation X'=AX then the solution of system of equation is…

Q: 29 What numbera in A-L makes A-QA@ possible (where Qis an orthognael matrix and A is an Zero or 1??

A: We shall solve this question by just using the definition of an orthogonal matrix. We know that a…

Q: independent

A:

Q: Let A be an n xn matrix and B= A+I, where I is the identity matrix, then it is possible for A and B…

A:

Q: Let A be an nxn matrix and B= A+I, where / is the identity matrix, then it is possible for A and B…

A:

Q: Which of the following matrices is positive definite?

A:

Q: Prove that if both products AB and BA are defined, then AB and BA are square matrices. Assume that A…

A: AB and BA are both defined.

Q: If A and B are invertible 2 × 2 matrices with nonnegative entries, then A + B must also be…

A: Let's take one counter example of the given conditions. That is A and B are invertible 2*2 matrices…

Q: Compute the matrix exponential e At for the following system x' = Ax. x'1 = 6x1 - 2x2, x'2 = 2x, +X2…

A: Step:-1 Note:- If A is diagonalizable matrix that is,  there exist an invertible matrix P and a…

Q: Suppose A, B, and C are three non-zero square matrices such that AC = BC. Is it true that A = B? If…

A: Not true

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.