4. Prove that thereal 2 × 2 matrix a b ^-[22] A d is invertible if and only if ad – bc, in which case find its inverse.

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: 5. For each of the statements given below decide if it is true or false. If it is true explain why.…

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Q: Ax = b has exactly one solution for every n*1 matrix b, then det(A) # 0

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Q: 2) An n x n matrix A is called nilpotent if Ak = 0 for some positive integer k. Show that if A + 0…

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Q: Let A, B, and A+B_ all be square invertible matrices of the same size. Show that (A +B')' = A(A+B)'…

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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: 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: Show that the matrices [1 1 1] A = |1 3 0 and B = |0 o 0 0 1 1 1 1 are similar.

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Q: uppose is the quadratic form Q = x Ax, where A is the symmetric matrix A = 2 he quadratic form is…

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Q: Suppose that the square matrix A is invertible. Show that the inverse of A is unique.

A: See below the explanation

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: 8 Consider a square matrix B whose inverse is given by B'. a In terms of B", what is the inverse of…

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

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Q: 9. Show that the matrices (6) 65), (8), and (1) generate M2×2(F).

A: We have to show that the matrix  and  generate

Q: TRUE or FALSE: Every invertible n xn matrix is similar to the identity matrix In.

A: To state true or false given statement.

Q: 34. Is it possible for A³ to be an identity matrix without A being invertible? Explain.

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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: Suppose A,B,C are each nxn invertible matrices. Show (ABC) CBA.

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Q: USE THE SAME MATRIX "E" from above. Verify E x inv(E) is a 4 x 4 Identity Matrix. Fill Up the Empty…

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Q: Show that a matrix B has a left inverse if and only if BT has a right inverse.

A: Let’s assume that C is a left inverse of matrix B i.e.

Q: 5. Show that a complex square matrix is unitary if and only if it is unitarily equivalent to a…

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Q: Let A be a 5 x 5 matrix such that each row of A contains every mumber from the following list…

A: From given matrix