Fill in the blank and explain why the following statement is true: "If A is an mxn matrix, then the columns of A are linearly independent if and only if A has____ pivot columns.

Q: Fill in the blank so that the resulting statement is true.

A: Given,                     A=abcd  , where  a, b, c, d are real numbers.       Remind that, any…

Q: A? 0 0 A =

A:

Q: Show that if the entries in each row of an n × n matrix A add up to zero, then the column vectors of…

A: linearly dependent means det of A is zero

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

A:

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: Determine whether the matrix is symmetric, skew-symmetric, or neither. A square matrix is…

A: Given matrix is

Q: In Exercise, find the singular values of the given matrix.

A: A singular value decomposition of is a factorization                             A = U∑V Twhere,…

Q: Write the third column of the matrix as a linear combination of the first two columns, if possible.…

A:

Q: Let A be an m×n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.

A:

Q: Find a basis of the image of the matrix below. 1 1 1 A = 125 1 3 7

A: Given matrix is A=111125137 We have to find the basis of the image of A. First of all reduce the…

Q: Let a be a real number, and consider the matrix a 2 1 1 6 3 A =

A: Given:

Q: In Exercise, find the singular values of the given matrix.

A: Given matrix will be as:- A = 1100   and   AT= 1010 Now,  ATA = 11001010 = 1+1000=2000 To Find the…

Q: Let x1 = (1/2, 1/2, 1/2, 1/2), x2 = (-2, –1, –4, –1), X3 = (1,2, –1, 1), x4 = (1, 1, 1, 0). (a) By…

A: Given that x1=12,12,12,12,x2=-2,-1,-4,-1,x3=1,2,-1,1,x4=1,1,1,0. Note: send part (b) and part (c)…

Q: Write the third column of the matrix as a linear combination of the first two columns, if possible.…

A:

Q: Show that if A is a square matrix that satisfies the equation A2 -2A +I= 0, then A-1 = 2I -A.

A:

Q: given A, Find a matrix P such that P-1AP = D, a diagonal matrix

A: The given matrix is: A=abba We have to find P such that: P-1AP=D, where D=A diagonal matrix.

Q: Use the concept of area of a parallelogram to write a statement about a 2 × 2 matrix A that is true…

A:

Q: Identify that the given matrix is Diagonalisable or not: -2 A = 2 6 -1 1 5.

A: It is diagonalisable

Q: Determine whether A is diagonalizable. If it is, find a nonsingular matrix P such that P−1AP is…

A:

Q: Let A be a 4x6 matrix. what is the dimension of Col A if the nullity is 2? Explain

A: 4

Q: Create a 5 x 3 matrix, call it A, with no more than 3 zeros such that the columns of A are linearly…

A: Let us consider a matrix A=-4-300-14103546-7-3-9 First we will show that columns of matrix are…

Q: let A nonsingular matrix, c #0, c scalar (cA)" = CA -1 -1 %3D Select one: True O False

A:

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

A:

Q: Write the third column of the matrix as a linear combination of the first two columns, if possible.…

A: A matrix is given. We need to check whether the third column can be written as a linear combination…

Q: Determine whether the matrix is symmetric, orthogonal, both, or neither.

A: Now transpose of A is, Hence A is a symmetric matrix.

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: In Exercise, find the singular values of the given matrix.

A: Find the product for the matrix A and transpose of A.…

Q: Determine if the matrix A = -1 1 1 is diagonalizable. If so, find P, D, so that -3 -1 -3 DDD-1

A:

Q: Use the Cayley-Hamilton theorem to find A1, A°, and A“ for the given matrix A. 10 -8 - 4

A:

Q: Write the third column of the matrix as a linear combination of the first two columns, if possible.…

A:

Q: Find all a,ß>0 so that the matrix 3 2 B [4]= a 5 B 2 1 a is strictly diagonally dominant.

A:

Q: Find the inserve if it exists, of the following matrix. 1 -1 2 1 1 1 1 2

A:

Q: The ith row vector of a matrix product AB can be computed by multiplying A by the ith row vector of…

A:

Q: Express the column matrix b as a linear combination of the celumns of 4. (Use A, A» an A = b =

A: We will use three colums of A as A1, A2, A3 and then solve it in step 2.

Q: Find a matrix B such that AB = I, where [ ] 8 1 1 0 A = and I = -8 8 0 1 1 a b Hint: Let B = %3D c d…

A:

Q: In Exercise, find the singular values of the given matrix.

A: We have the given matrix A as A=34 Now, transpose of matrix A is calculated as AT=34 Now, the value…

Q: Show that if A is a square matrix that satisfies the equation A2 - 2A + I = O, then A-1 = 2I - A.

A: Given that,                    A2-2A+I=0

Q: Fill in the missing entries of Q to make Q an orthogonal matrix

A:

Q: In Exercise, find the singular values of the given matrix.

A: Consider  A=2102. Now ATA=20122102=2223

Q: If A is a square matrix such that A3 = O, show that (I-A)~l = I + A + A2.

A: To prove that (I-A)~l = I + A + A2 If A is a square matrix such that A3 = O

Q: Use the definition of Ax to write the vector equation as a matrix equation -2 + Z2 - 5 3 +Z4 2 4 Z1…

A:

Q: Determine whether the matrix is symmetric, orthogonal, both, or neither.

A: Now transpose of matrix A is, Therefore A is a symmetric matrix.

Q: Let A be a 4 × 5 matrix. If a1, a2, and a4 are linearly independent and a3 = a1 + 2a2, a5 = 2a1 − a2…

A:

Q: Write the third column of the matrix as a linear combination of the first two columns, if possible.…

A: consider the equation as follows  a158+b249=5720a5a8a+2b4b9b=5720a+2b5a+4b8a+9b=5720…

Q: Find the maximum number of linearly independent rows in the following matrix

A: Given matrix - z = 01-3-1001131110-220Number of linearly independent rows in a matrix is called as…

Q: Find a non-zero vector V in the row space of the matrix -20 4 0 -9 -3 12 -5 0 25 -12 -4 16 V

A:

Q: State whether the matrix is diagonalizable or not. [:] 0 7 0 0 O diagonalizable O not diagonalizable

A: Given matrix is 0700 and now subtract λ from the diagonal elements=0-λ700-λThe determinant is…

Q: Express the column matrix b as a linear combination of the columns of A. (Use A,, A,, and A,…

A: The solution is solved by applying Gauss- Jordan elimination method, for (equation) linear…

Q: Find c such that the number of the pivots to the matrix below is 2: 1 1 1 -1

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.