Let A be an n x n matrix such that Am = 0 for some positive integer m. Show that A is not invertible. Give a full proof.

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: Let S be an n × n singular matrix. Prove that for any n × n matrix B, the matrix SB is also…

A: Singular matrix :A matrix is singular if and only if it has a determinant of 0

Q: Given a square matrix A, prove that A is invertible if and only if ATA is invertible.

A:

Q: Prove that if A is an n × n invertible matrix, then adj(A−1) = [adj(A)]−1.

A: By definition of A-1, we have

Q: Prove that for any m x n matrix A, rank(A) = 0 if and only if A is the zero matrix.

A: Given, A is m x n matrix To prove:

Q: Use the Invertible Matrix Theorem to decide if A is invertible

A:

Q: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

A: Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP…

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: Algbra prove that A is invertible if AT(a transpose)+A is a positive definite matrix.

A: Given that If A^T+A is a positive definite matrix For a positive definite matrix Det(A^T+A)>0. So…

Q: Let A and B be n×n matrices such that AB is invertible. Prove that A and B are invertible. Give an…

A: It is given that AB is invertible, that means there exist some matrix C such that:(AB)C = C(AB) = I…

Q: Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

A:

Q: Prove that if an n × n matrix A is not invertible, then A[adj(A)] is the zero matrix.

A: 1). Matrix A is invertible if and only if matrix A is a non singular matrix. That is, determinant of…

Q: Let A be a nonsingular matrix. Show that det(A−1) = 1/det(A)

A:

Q: Find the matrix C such that BC = AT

A: Let A be an n×n matrix, Then the transpose matrix of A is also an n×n matrix denoted by AT. Now,…

Q: Let A be an m x n matrix. Prove that if c is any nonzero scalar, thenrank(cA) = rank(A).

A:

Q: Let A be an m × n matrix with rank m. Prove that there exists a n × m matrix B such that AB = Im:

A: Take any n x m matrix B with rank n. We know that rank AB = rank A = m (for n>m) hence AB is…

Q: Assume that A is an odd order skew symmetric matrix. Prove that det(·) is an odd function in this…

A:

Q: Let A and B be n × n matrices such that AB is singular. Prove that either A or B is singular.

A: Let A and B be n × n matrices such that AB is singular.

Q: Let A be an nxn matrix. Prove that dim(col(A²)) < dim(col(A)).

A: Given: Assume that, A is an n×n matrix.

Q: Prove that the n X n identity matrix In is invertible and that

A: We have to prove that the n X n identity matrix In is invertible and that In-1=In.

Q: Prove If A is an invertible matrix, then AT is invertible and (AT)-1=(A-1)T

A:

Q: Suppose A E M,xn(C) is a normal, invertible matrix. Prove that A A- is unitary.

A: Given A € Mnxn(C) is a normal matrix.

Q: Let A be an m × n matrix. Prove that if AT A is nonsingu- lar, then rank A = n.

A: Given that, A is an n x n matrix. A^T is the transpose matrix of A. AT A is non-singular. We have to…

Q: If A is an n × n matrix, show that (Ak )T = (AT )k for any positive integer k.

A: Matrix: We will solve it by the method of induction:(Ak)T=(AT)k case 1: k=1case 2: k=ncase 3: k=n+1

Q: Let A be an m ×n matrix with rank m. Prove that there exists an n ×m matrix B such that AB = Im.

A: Let A be a mxn matrix with rank m. To prove that there exist a nxm matrix B such that AB = Im.

Q: let A and B be n ×n invertible matrices. Prove that AB invertible and (AB)−1 = B−1A−1.

A: Let A and B be n×n invertible matrices, with inverses A–1 and B–1, respectively. Consider their…

Q: Let A be an Hermitian matrix and let B = iA. Show that B is skew Hermitian.

A:

Q: Let A be an n × n matrix. Prove that if Ax = 0 for all n × 1 matrices x, then A = O .

A: we have been given that Ax = 0 for all n × 1 matrix x and  A be an n × n matrix.

Q: Show that if A is a symmetric positive definite matrix, then A is nonsingular and A−1 is also…

A:

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: If A be an n x n invertible matrix. Prove that the inverse of A is unique.

A: The assumption for the matrices helps to prove the required. A matrix is said to be invertible if…

Q: Let a be a real number. Let 1 1 A = | 0 1 1 1 1 2 a) Find the value of a for which the matrix is not…

A:

Q: Show that if A is an m × n matrix such that AAT is non- singular, then rank A = m.

A: Let AAT can be non singular only if the columns in A are linear independent and they span the column…

Q: Prove that the n X n identity matrix In is invertible and that ln-l = In

A:

Q: Let A = CD, where C and D are n × n matrices and C is invertible. Prove that the matrix product DC…

A: A = CD, where C and D are n × n matrices, and C is invertible. We need to Prove that the matrix…

Q: Assume that A and B are both n x n matrix. If|AB| + 0, then both A and Bare invertible. O True O…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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: Let A E Mn(C) denote the n x n matrix with all entries equal to 1. Show that A is positive…

A: Let A belongs to  Mn (C) denote the n x n matrix with all entries equal to 1. Show that A is…

Q: Suppose A is an n×n invertible matrix such that A – 2A = I, . Find an for A. expression

A:

Q: Find a non-zero 2 × 2 matrix A such that A2 = A =

A: We have to find a A matrix of 2x2 such that, A2  = 0000  So let suppose a matrix    Satisfy the…

Q: Prove that A+ A' is a symmetric matrix, w

A:

Q: Show that if an n × n matrix A is positive definite, then there exists a positive definite matrix B…

A: Consider the provided question,

Q: Let A be an m × n matrix. Show that ATA and AAT are both symmetric.

A: Given A be an m×n matrix. we have to show that ATA and AAT are both symmetric.we will make use of…

Q: Prove that if A is an invertible matrix and B is row equivalentto A, then B is also invertible.

A: Recall the following equivalent statements as follows.

Q: Let A be a nilpotent matrix (that is, Am = 0 for some m &gt; 1). Prove that if A is diagonalizable,…

A: Consider a nilpotent matrix A. So Am=0 for some m&gt;1.

Q: Let B be an n x m matrix with rank m. Prove that there exists an m x n matrix A such that AB = Im.

A: Consider the given matrix n×m with rank m

Q: Let A be an m × n matrix.(a) Describe the diagonal entries of AT A in terms of the columns of A.(b)…

A: (a) Describe the diagonal entries of AT A in terms of the columns of A.

Q: Find x such that the matrix is singular.

A:

Q: - L

A: Given that A=42-12 Take  A-λI=04-λ2-12-λ=0(4-λ)(2-λ)+2=08-6λ+λ2+2=0λ2-6λ+10=0λ=6±36-402λ=6±2i2λ=3±i

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.