35. a. Consider the upper triangular 3 x 3 matrix a b A = |0 d 0 f] e For which values of a, b, c, d, e, and f is A invert- ible? b. More generally, when is an upper triangular matrix (of arbitrary size) invertible? c. If an upper triangular matrix is invertible, is its in- verse an upper triangular matrix as well?

Q: Suppose that A is a 18 x 7 matrix and the null space of A has dimension 3. What is the rank of A?

A: For a m×n matrix, the number of rows is equal to m and the number of column is equal to n .

Q: 3. Let A be an n x n singular matrix. Describe one way to construct a non-zero matrix B such that AB…

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Q: Let = Find a nonsingular matrix P and a diagonal matrix D such that P-'AP = D.

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Q: 4. Let A be a square matrix with transpose A". • If AT = A, then A is called a symmetric matrix. %3D…

A: If A is non singular then A is non zero. ATT = A

Q: 65. Final exam, Winter 2015. A matrix A is said to be skew-symmetric if AT = -A. For instance 03 М-…

A: 65.(a) A is skew symmetric matrix then AT=-A Using the results: (1) det(AT)=det(A) (2) det(cA)=cn…

Q: 2. Suppose that A is a 64 × 17 matrix and that the rank of A is 11. What is the dimension of the…

A: The given matrix is A of size 64×17. Also the rank of A is 11. As we know that for any matrix of…

Q: 3. Let A = 3-1 1 V₁ = 3 -2 2 = 2 V₂ = 3 V3 = 6 V4 and use the facts Av₁ = V₁ AV₂ = V₂ AV3 = V3 V4 €…

A: 3. Let , and use the facts  to answer the following question.(c) If A is diagonalizable then give…

Q: 6. If A is an invertible matrix, show that its projection matrix, P, is the identity matrix, I.

A:

Q: 5. (a) True of False(and why?): for any n x n matrices A,B we have rank(AB) rank(BA)? = (b) Let A be…

A: For false statement, we will provide a counterexample.

Q: How do we know that the ith row of an invertible matrix B is orthogonal to the jth column of B if i…

A: We have been given a matrix B. To Prove: ith row of B and jth col of B-1 are orthogonal, where B 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…

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Q: . Can a square matrix with two identical columns be invertible? Prove it clearly.

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Q: 5. Let A = 7453 7-9 -4 5 3-3 -7 6 7 -2 -6 -5 5 8 5-6 2 8-1-7-4 8 449 3 -4 5 -7 -6 -3 5 6-8-5 •…

A: Given A=7-9-453-3-7-467-2-6-555-7-65-628-358-1-7-486-8-54493. Applying row operations: Apply R1→R17:…

Q: 3. Diagonalize the following matrix. Use this diagonalization to compute A¹⁰ 1 А A = [¹₁2] 2.

A:

Q: 3.) Let Find A1 A- 203 010 100 using confactor representation of the imorse matrixe

A: The given problem is to find the inverse of the given matrix by using the cofactor representation of…

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: Using the Rank-Nullity Theorem, explain why an n x n matrix A will not be invertible if rank(A) < n.

A: The Rank-Nullity Theorem states that for a linear transformation T:V→W between finite-dimensional…

Q: Prove that the product of two n x n anti-diagonal matrices is diagonal.

A: To prove that the product of two n*n anti-diagonal matrices is a diagonal matrix.

Q: b) Let A, B, and C be nxn matrices and suppose that ABC is invertible. Which of A, B, and C are…

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Q: Let A E Rx and B E Rmxm and assume that dim N(A) = k and dim N(B) = l. Let C be an 2 x 2 block…

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Q: Let A = UEVT be a SVD of a square matrix A. What is the product of the diagonal elements of E, in…

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Q: 3. Suppose that A is an n x n matrix. a. Suppose that A? = AA is invertible with inverse B. This…

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Q: 3. Suppose A is an n x n matrix such that A³-3A² +21= Onn. Show that A is invertible and find its…

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Q: 6. Prove that the matrix A = -1 2 is diagonalizable, and find the corresponding diagonal matrix D…

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