Prove that: Det AB = Det A x Det B where matrix A and matrix B are: A = [ 1 -2 3          -2 3 1          0 1 0 ] B = [ 1 0 2          3 -2 5          2 1 3 ]

Q: - ; Let A be a square matrix such that A? – 7A+ 81 = 0 (where I and 0 are the identity and zero…

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Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: Find two 3 x 3 matrices A and B such that det(A + B) + det(A) + det(B)

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A: For a n×n matrix trace of matrix is the sum of the diagonal elements. Let the matrix A be…

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- BT) =…

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Q: Let A and B be 2 x 2 matrices with real entries. Determine if the following statements are true or…

A: We will find out the required solution.

Q: Let A be an n X n matrix with n > 1 and suppose A satisfies Ą² - Which one of the following…

A: True of false statement

Q: Let A and B be 2 x 2 matrices. Prove that det(AB) det (A) det(B)

A: To prove that  det(AB)=det(A)det(B)

Q: Let A and B be n × n matrices. Prove that if A is nonsingular, then AB is similar to BA.

A: It is given that matrices  A and B are of order (n x n). To show that AB is similar to BA. Since,…

Q: Use the matrix P to show that the matrices A and A′ are similar.

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Q: Show that if A is a nonsingular matrix, then adj A is non- singular and 1 (adj A)- = A = adj(A-).…

A: Given: A is a nonsingular matrix. Since A≠0

Q: (BH) Consider the following matrix in R5×5. * 0 0 0 0 0 0 0 0 0 * A = * * * where the *s are…

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

A: Here, A and B be two 3×3 matrices such that det(A)=2 and det(B)=3. We need to find the value of…

Q: Let A be a nonsingular matrix. Show that det(A−1) = 1/det(A)

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Q: Let n > 2 and let A be an n xn matrix. Prove that "det(adj A) = det(A)"-1."

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Q: Consider the matrix [1 2 3] B = |2 3 4 3 4 6 (a) Show, by calculating det B, that B has an inverse…

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Q: Let A be a 3 x 3 matrix such that det(A) = 2. Evaluate det(Aª) and det(2A).

A: Recall the following. If A is a n×n matrix and m be any positive scalardet k A=kndet Adet Am=det Am

Q: Let A and B be n x n matrices such that AB=In. Prove A=B-1 (and hence B=   A-1).

A: Given: -

Q: Prove that, for m X n matrices A and B, rank (A + B) <=rank(A) + rank(B

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Q: If A and B are matrices such that AB is defined, then it is true that (AB)T = ATBT %3D True False

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Q: b) Prove: Let A be an m x n matrix, R" = NS(A) + RS(A). %3D

A: A be an m×n matrix, we have to prove that ℝn=NSA⊕RSA  …1.

Q: If A is a skew-symmetric n X n matrix and n is odd, prove that det A = 0.

A: Since A is a skew-symmetric matrix we have,  AT=-A where AT is the transpose of matrix A. Then we…

Q: Prove: If A and B are m × n matrices such that Ax = Bx for all x ∈ R n , then A = B.

A: Given, If A and B are m×n matrices such that Ax=Bx for all x∈Rn Prove that: A=B

Q: 1. Prove the following for any two matrices A and B of order ( 2 × 2 ). (A + B)² = A² + AB + BA + B2…

A: Here (a) and (b) are unrelated problems. These 2 are unrelated problems as anyone can be solve…

Q: If  det(A) =|A|= 0, then A is called a non-singular matrix  True  False

A: Given statement is false because

Q: Let A and B be 2 x 2 matrices. Prove that det(AB) = det(A) det(B)

A: See the attachment

Q: Find a 3 x 3 matrix A such that A A A A = 0 0 = 2 5 0 3 and

A: Given A100=-1-41,  A010=251  and A001=003

Q: Let A be a 2 x 2 matrix. If det (A) # 0, then det (adj A) = det (A).

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Q: If A and B are square matrices of the same size such that det (A) = det(B), then det(A+ B) =…

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Q: If A is an orthogonal matrix, then det(A) = ±1.

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Q: Calculate adj A, det A, and hence A¯', for the matrix A=| 2 1 -2 -1 1

A: To find the adjoint determinant and inverse of a matrix

Q: Let A and B be two matrices of size 3 x3 such that det(A) = -2 and det(B) = 2. Then det(2A 2B") =

A: the solution is following.

Q: Find a 3 x 3 matrix A such that A A A A = 1 4 0 = 2 0 -1 B-B EL. 0 2 1 = -3 and 0 -5 0 -5 B-B 0 -5 1…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

Q: 1. Prove or disprove. a) det(A + B) = det(A) + det(B) b) det(AB"A')= det B c) (A+B)(A – B) = A² –…

A: According to the policy, I am doing the first three subparts. Explanation of the answer is as…

Q: Find the value of k such that matrix A is singular: (1 2 A = | 5 -1 2 6. k.

A: Singular Matrix: A square matrix that is not invertible is called singular or degenerate i.e., it is…

Q: [1 0 0 41 250 3 3 00 2 L4 3 2 1 ) Let, A = be a 4x4 matrix. Find det (A) and det (2ATA).

A: To find |A|   Also to find |2AT A|

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: Find 2 x 2 matrices A and B such that det(A + B) det A + det B.

A: We have to find matrices A and B such that det(A + B) ≠ det A + det B

Q: Prove that if A and B are matrices such that A is symmetric, then (BA−1)T(A-1BT)-1=In

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Q: Let A and B be 3 x 3 matrices, such that det(A) = 2, det(B) = -4, then det(2AT B-2) = %3D %3D A) 1…

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Q: For a non-singular matrix A, A^(-1) is equal to: (adj(A)) / det(A) det(A) x (adj(A)) det(A) x A

A: Explained below

Q: Prove that if A is an m x n matrix, B is an n x p matrix and C is a p x q matrix, then (AB)C= A(BC).

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Q: (a) (MP) Verify that det(AB) = (det A)(det B) for the following matrices: 1 -2 3 1 0 2 A = -2 3 1 B…

A: given matrix  how to find determinant

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

A: A and B are two matrices of size 3×3 such that detA=-2 and detB=3

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

A: If A and B are n x n non-singular matrices then prove that (AB)-1=B-1A^-1

Q: Prove that if ∣A∣ = ∣B∣ ≠ 0, and A and B are of the same size, then there exists a matrix C such…

A: we have |A|=|B|≠0   A and B are of the same size    We have to prove |C|= and A=CB

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: 2. Let A be a 3× 5 matrix such that A = [1 2 ūz v4 05), where ī1, i2, 03, 04, ūz E R³, and Find ūz…

A: We have to solve given problem: