Let p and q be nonnegative integers and let A be a square matrix. Show that Ap Aq = Ap+q and (Ap ) q = Apq .

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

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Q: If A and B are n × n diagonal matrices, show that AB = BA.

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Q: For which values of the constants b and c is the follow- ing matrix invertible? 1 b -1 -b

A: The given matrix is 01b-10c-b-c0

Q: For a nonsingular matrix A and a nonnegative integer p, show that (Ap)−1 = (A−1)p.

A: The objective is to show that AP-1=A-1P for a non singular matrix A and a non negative integer P

Q: Express the quadratic form (c₁r1 + c₂2+...+Cnn)² in the matrix notation TA, where A is symmetric.

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Q: Find conditions on a, b, c, and d such that B = as attached commutes with every 2X2 matrix

A: B=abcd Let A=uvxw BA=au+bxav+bwcu+dxcv+dwAB=ua+vcub+vdxa+wcxb+wd Let BA=AB…

Q: Let A and B be nxn inversible matrix. Prove that AB is invertible and (AB)-1=  B-1A-1.

A: Let A and B be n x n invertible matrices, with inverses A-1 and B-1, respectively.

Q: Suppose that  |A| + |B| = |A + B| , find a 2x2 matrix for A and B

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Q: Show that AB and BA are not equal for the matrices

A: Given: The matrices are given as, Finding the value of AB by using matrix multiplication,

Q: Give an example of a nonzero 2 X 2 matrix A such that A2=O

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Q: Find the symmetric matrix A associated with the given quadratic form. x12+ 2x22 + 6x1x2

A: Given: The symmetric matrix A for the general quadratic form is written as follows, By comparing…

Q: let Q(x)= x12 -8x1x2 -5x22 , then matrix A of the quadratic from is A=?

A: Explanation is given below..

Q: If A is a 31x24 matrix whose rank equals 10, find dim(Nul(AT)). Your Answer:

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Q: If Ax = ax for nxn matrix A, nx1 matrix x, and a E R, determine a scalar 3 with the property that…

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Q: Show that any nX n matrix can be written as the sum of a symmetric matrix and a skew-symmetric…

A: To show that any n x n matrix can be written as the sum of a symmetric matrix and a skew-symmetric…

Q: Let A be an n × n matrix in which the entries of each row sum to zero. Find ∣A∣.

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Q: Let If A is 2x2 matrix and /(4)=0, then %3D

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Q: Suppose that A+ is a 2 × 2 matrix and that A+ = -A. Show 7.2 that iA is awesome.

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Q: Find a 2 x 2 matrix such that -3 1

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Q: Find a 2 x 2 matrix B so that B2 = A, where

A: Given matrix A=-7-16817 To find matrix B such that B2=A.

Q: Let A and B be square matrices of same order. Prov nomials of AB and BA are same. Do AB and BA have

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Q: Give the matrix associated with the quadratic form 10(x,)? – 5x,x2 +6(x,)².

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Q: Let p(x) = x³ – 2x + 4 and compute p(A). - -- A = -3 2 4 NOTE: Write the elements of the matrix…

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Q: Find the matrix corresponding to the quadratic form Q(x) = x + 2x1x2 + 2x1x3. Is Q positive…

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Q: Let A be a square matrix and det(A) = 2. If B is the matrix that results when –7 times the - first…

A: Given that: det(A)=2

Q: Let p(x) = x³ – 2x + 4 and compute p(A). |2 A = |0 -3 4 NOTE: Write the elements of the matrix…

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Q: Find nonzero 2 × 2 matrices A and B such that AB = O.

A: Consider

Q: Construct a 3 × 2 nonzero matrix A such that: Ax = 0 has only the trivial solution.

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Q: Find the symmetric matrix A associated with the given quadratic form. x1x2

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Q: Let A and B be square matrices of order 3 such that |A-| = x + 7 with x + -7 and |A * A" | = . What…

A: A and B are square matrices. Therefore, det(AB)=det(A).det(B) det(A)=det((A)T) We will use these…

Q: Find a non-zero 2 × 2 matrix such that -5 -1 25

A: Let a 2×2 matrix be: abcd such that: -5-1255 abcd=0000

Q: Show by direct calculation for 2x2 matrices A, B, C that A(BC)

A: To prove that  for 2×2 matrices A, B,  that A(BC)=(AB)C.

Q: Suppose that Q is a matrix with the property Q−1 =QT . Show thatqij = Qij/det(Q)

A: Given: Q-1 = QT To Prove: qij = Qij/det(Q) Formula Used: If A is a square matrix, then its inverse…

Q: Let A be an n x n matrix and let D be a n x n diagonal matrix. Show that AD =…

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Q: If A is a positive definite matrix and B is positive semi-definite, show that trace (AB) 2 O with…

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Q: Find the nullspace of the matrix

A: The nullspace of the matrix is to be determined, The matrix is shown below as,

Q: Find the matrix B such that 2AT+3B=0.

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Q: Find two unequal 2 x 2 matrices A and B such that 1 01 АВ

A: Given: AB=1001

Q: Let A be an n × n matrix and c a real number. Show that if cA = O , then c = 0 or A = O .

A: Given: A be an n × n matrix and c be a real number. To prove that: if cA=0 , then c=0 or A=0.

Q: Find two nonzero matrices A and B such that AB= 0

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Q: Find matrix C

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Q: Find a non-zero 2 × 2 matrix such that -5 -1 0 0 25 ||

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Q: Show that if B is a symmetric nonsingular matrix, then B2 is positive definite.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Find the 2 x 2 matrix A, if 2 (A" – 212x2)-

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Q: Let A be a 4x4 matrix and det(A)=2. Find det(2A-)'

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Q: Please show how to determine whether matrix A is diagonalizable: If possible, find an invertible P…

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Q: Show that A is similar to B by showing that A and B are similar to the same diagonal matrix, where…

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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.