consider the following definitions: 1. a submatrix of an matrix A is any smaller matrix obtained by removing one or more rows and / or columns of A 2. the rank of a matrix Amxn can be defined as the largest integer p for which some of submatrix p x p of A has non-zero determinant. If A is square, with det(A) ≠ 0, the opposite of A is the number of lines (or columns). Using just this definition of rank and properties of the determinants to simplify the calculations without scaling the matrix determine all possible values ​​of the parameter t, for matrix A, then have rank 2. Show mathematical argument and justify your answer

Q: Suppose A is n x n invertible matrix ( AA-1 = I), then the first column of A-1 is orthogonal to the…

A:

Q: (a) det(A-1) (b) det(5A)

A:

Q: Let M be a 10 × 10 real matrix such that M2 = M and the determinant of the matrix cannot be 1. Does…

A:

Q: 1 Suppose that a 3 x 3 matrix A has det A = 1 and that its cofactor matrix is C = 1 2 |-7 4 2 Find…

A: Formula: Let A be a n×n matrix with non zero determinant then, A-1=1det ACT where C is the cofactor…

Q: Consider an 5 x 5 matrix A satisfying the following two properties: (a) Every row has exactly one…

A:

Q: Solve the following system by applying the Cramer's rule and find the determinant of matrix A (det…

A:

Q: For which value of x is the following matrix not invertible: 1 1 3x A = 3x 3x 3x 3x 3x O 2 4 O1/3 1

A:

Q: The determinate of Matrix A is: * 3-2 A = [1 4 0.5] 2 3 O 16.5 O 1.5 O 11 Non

A:

Q: Given that A, B are two 4x4 matrices such that A B = 1, where l is a 4x4 identity matrix, which of…

A:

Q: If you know all 16 cofactors of a 4 by 4 invertible matrix A, how would you find A?

A: Given, an invertible square matrix A of order 4, and we know all the 16 cofactors of this matrix.…

Q: Given the following Matrices: Find the determinant of [3(BT+2A)] * (AB-1) using the Pivotal Method.

A:

Q: 20. Let A = (a;j) be a given (3 × 3) matrix. Form the associated (3 x 5) matrix B shown next: a12 B…

A:

Q: Evaluate the determinant of matrix det(A), where 1 1 1 ... X2 T3 In ... A = ... n-1 „n-1

A:

Q: 8. (BH) Consider the following matrix: mi1 m12 m13 M = | m21 m22 m23 m21 m22 m23 (a) Show directly…

A:

Q: If the determinant of a 5 x 5 matrix A is det(A) = –10, and the matrix C is obtained from A by…

A:

Q: Given a 3x matrix in terms of scalar c and that det (A)=8, find the valuddof G if -1 1c-2 C+7 3. 2.

A:

Q: If the determinant of a 5×5 matrix A is det(A)=8, and the matrix C is obtained from A by swapping…

A: The determinant of a 5×5 matrix A is detA = 8 Formula Used: detC = -detA To evaluate: The value of…

Q: A 3 x 3 matrix M is given as (x 1 x-1 M 16. =2 1 1 x zero, then A 2x2 - 3x - 1 = 0 B 2x2 + 3x + 1 =…

A:

Q: Show that if A is any m × n matrix, then AAT and AT A are symmetric.

A: There is a matrix  A given of order m×n. To show that AAT  and  ATA  are symmetric. Solution: The…

Q: For which value of x is the following matrix not invertible: 1 1 2x A = 1 2x 2x 2x 2x 2x 4 1 -1 1 2

A: For a matrix A to not be invertible: dat(A) = 0.

Q: 2 Consider the 3 x 3 matrix B = 1 2h -1 1 1

A:

Q: -1 1 2 1 2 3 1 2 3 -1 2 3

A: Given lower triangular matrix :100⋯0120⋯0123⋯0123⋱0123⋯n

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: Solve the following system by applying the Cramer's rule and find the determinant of matrix A (det…

A:

Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

A:

Q: Given the following Matrices: Find the determinant of [3(BT+2A)] * (AB-1) using the Pivotal Method.…

A:

Q: If A is a 5x5 invertible matrix, then the determinant of 2A is k|A| where k is equal to A 16 B 0 2…

A:

Q: For each of the following matrices, compute the determinants of all the leading principal…

A:

Q: 2. If A is a 3 x 3 matrix, prove that the determinant of A is equal to the determinant of AT.

A: 2. Given A is a 3×3 matrix we need to show that determinant of A is equal to AT. Let A=abcdefghi.…

Q: For a nonsingular matrix A and nonzero scalar k, show that (k A)−1 = 1 k A−1.

A: Given that A is nonsingular matrix. That is, A≠0. Since k is a non zero scalar, kA≠0. Therefore, kA…

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

A: Consider a skew symmetric n x n matrix A where n is odd. Here the objective is to prove that det(A)…

Q: Consider the following matrix 3 -3 A = 3 4 -4 2 -3 a. Compute det(A) by using the special pattern…

A: We will use row operation to find row echelon form of matrix.

Q: Consider the NxN-dimensional matrices A and B, where B=S¹AS (with S an NXN-dimensional non- singular…

A:

Q: Determine whether the following matrix is invertible by finding its determinant. 4 -14 -56

A: Consider the matrix 14-14-56

Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…

A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…

Q: Let A is an orthogonal matrix, that is, AAT = I and B? = I. Compute the determinanm of the matrix…

A:

Q: Let A be a symmetric 2 X 2 matrix and let k be a scalar. Prove that the graph of the quadratic…

A:

Q: Explain why the columns of an nxn matrix A span R" when A is invertible. Choose the correct answer…

A:

Q: If the determinant of a 4×4 matrix A is det(A)=2, and the matrix D is obtained from A by adding 5…

A: Given det A= 2  D can be obtained by adding 5 times the third row to the second in A  See the…

Q: If the determinant of a 4 x 4 matrix A is det(A) = 7, and the matrix B is obtained from A by…

A:

Q: Determine whether each of the following matrices is positive definite. If it is not, give a…

A:

Q: Consider the following 3 x 3 system. [an a12 a13 a21 a22 a23 a31 а32 аз3 Use elementary row…

A:

Q: Suppose AB = AC, where B and C are nxp matrices and A is invertible. Show that B = C. Is this true,…

A: Given information: Consider AB=AC and B and C are n×p matrices and A is invertible.  Calculation:…

Q: Let A be an x n matrix and let det(A) denote its determinant. Then O the null space of A has…

A:

Q: [1]a- Prove that a square matrix A is invertible iff A"A is invertible . b- Prove that without…

A:

Q: Consider the following matrix 3 1 -2 A = -1 1 1 a) By evaluating the determinant of A determine in…

A:

Q: 12. Use the properties of the determinant function to show that if A and B are any n × n matrices,…

A:

Q: Consider the following matrix 3 1 -2 A -1 1 1 1 0 a) By evaluating the determinant of A determine in…

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.