2. Let a, b be non-zero real numbers. Let T be an n × n matrix. If T satisfies (Tal₁)(T - bln) = Onn show that 1 T-¹=(-T+(a + b)In). ab In your proof, you must explicitly state any properties or theorems you have used.

Q: Q: Determine if the following system is consistant q using matrix ? X2 - 4X3 2X1 -3X2 +2X3 =l 5x,…

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Q: A B Let m and n be whole numbers. Matrix P is of order/size m x n. Matrix Q is of order/size n x m.…

A: Given that, m and n are whole numbers. Matrix P is of order m×n and  matrix Q is of order n×m. Want:…

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Q: Verify Cayley-Hamilton theorem for the matrix A = and find its inverse. Also express A - 4A" 2 3 -…

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Q: Example: f=(f¹, f²): R³ → R² (-6) T1I2X3 that is, f¹(x) = x₁²² and ƒ²(x) = 1₁T2Tz. Then the Jacobi…

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Q: If A is an nxn matrix, then the equation Ax = b has at least one solution for each b in R".

A: (C)

Q: 6. Find the matrix of the quadratic form. Assume x is in R³. a. 3x} – 2x3 + 5.x + 4x1x2 – 6x1X3 b.…

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Q: Show that the matrix X -1 : -1 -1 X : 0 : : X is Positive Definite if x > √n −1 The lower right…

A: The given matrix is An=x-1...-1-1x00::⋱:-100x. To Prove: This matrix is positive definite if…

Q: If A²=0, prove that A cannot be invertible.

A: Given,If A²=0, prove that A cannot be invertible.

Q: We call g(x1, x₂) = ax² + 2bx1x2 + cx a quadratic form, a, b, c are constants. Suppose X1 X = try to…

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Q: Let 2 -1 | -1 2 -1 -1 -1 -1 -1 –1 -1 2 -1 2 be the Eç Cartan matrix corresponding to the simple…

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Q: 13.) Determine the matrix: s P 6-8 3 ។ -3 the dimension X 10

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Q: Q9. As usual, let In denote the n x n identity matrix. 1 - (a) Find a 2 × 2 matrix such that A² =…

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Q: Suppose I + AB is invertible. Show that I + BA is also invertible by showing that (I + BA)¯ = I –…

A: Assume that I+AB is invertible.  Then, I+ABI+AB-1=I ⋯ 1. Use equation (1) and calculate…

Q: EXER 7: MATRICES Prove that if N is invertible then MN¹ = N¹M if and only if MN=NM.

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Q: Exercise 51. Is there a square matrix X with real entries such that det(X¹X) = -3?

A: detXTX=-3 X is a square matrix with real entries.

Q: Show that any positive definite square root of a diagonal, positive definite 2 x 2 matrix must also…

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Q: Please write clearly. Thank you

A: a) To find the matrix A corresponding to the quadratic form: We need to identify the coefficients of…

Q: Which of the following statements is/are correct? Please select one or more: For every matrix A,…

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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 v4 = 20 – 02 + 503. Let w = 601 + 202 – 303 – 204. (a) Write w as a linear combination of…

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Q: Write f= x + 10x1x2 + x3 as a difference of squares, and f= x? + 10x,x2 + 30x as a sum of squares.…

A: Given f=x12+10x1x2+x22.  Let the associated matrix of f be A=abba is 2×2 symmetric matrix. Then,…

Q: For each 2 x 2 real matrix A find the canonical form B and an invertible matrix P such that P-1 AP =…

A: A square matrix A of order 2x2 is said to be diagonalizable if there exist an invertiable matrix P…

Q: diag(a1, ..., an) be a diagonal matrix (i.e., the matrix with the numbers a1, ..., an on the main…

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Q: For each 2 x 2 real matrix A find the canonical form B and an invertible matrix P such that P-1AP =…

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Q: For each of the following statements, indicate whether it is true or false. 2. justification is…

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Q: 3. Find m and n so that the matrix below is symmetric:

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Q: suppose an nxn matrix has positive diagonal entries and negative everywhere else. suppose sum of the…

A: Introduction: To prove that the determinant of matrix A is non-zero, we will show that A is…

Q: nts) Check the definiteness of the following matrices A and B; -1 1 1] A = |1 5 L3 0 5 0 B = 3 -1 2…

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Q: Suppose that B = (CEED-·B·CD) -5 and C= 0 are ordered bases of R³. Find the conversion matrix from B…

A: Concept: A vector r is said to be a linear combination of vectors a ,b and c.. etc., if there exist…

Q: The equation A² - 2A + I = O implies that ---Select--- Need Help? Read It ✓. It follows that…

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Q: b) Evaluate det[(2A)-1 (3A")], where A is any invertible (2 x 2) matrix.

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Q: o matrices A and B one has that Null(B) is contained in Null(A · B).

A: Given, for two matrices A and B, to show that Null (B) contained in Null(A·B).

Q: Compute u + v, 2u – v, 3u – 2v, and 0 – 3v if (a) u =| 2 3 2 (b) u= = -3 (c) u = 1 4

A: (a) Consider the given matrices, u=123 and v=201 now, use the matrices addition property,…

Q: If a 6 x 6 matrix contains 19 zeros, show that there are 4 zeros which are all in distinct rows and…

A: A 6x6 matrix has 36 entries. Given if a 6x6 matrix contains 19 zeros, then to show that there are 4…