Lin Alg Multiple Choice- Test 3

let A and B be arbitrary matrices. each column of AB is a linear combination of the columns of B using weights from the corresponding column of A

false. rows of A

the second row of AB is the second row of A multiplied on the right by B

true

the transpose of a product of matrices equals the product of their transposes in the same order

false. (AB)^T=B^T*A^T

suppose the last column of AB is entirely zero by B itself has no column of zeroes. what can you say about the columns of A?

homogeneous linear system has nontrivial solution, column vectors of coefficient matrix linearly dependent

if A= [a,b c,d] and ad=bc, then A is not invertible

true

each elementary matrix is invertible

true

if A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In

false. row operation transforms In →A^-1

need to memorize!: let A be a square nxn matrix, then the following statements are equivalent.

A is an invertible, A is row equivalent to nxn identity matrix, A has n pivot points, the equation Ax=0 has only trivial solution, the columns of A form a linearly independent set, the equation of Ax=b has a unique solution for each b in R^n, the columns of A span R^n, there is an nxn matrix C such that CA=I, there is an nxn matrix D such that AD=I, A^T is an invertible matrix

if the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix

true

if the columns of A are linearly independent, then the columns of A span R^n

true

if A is an nxn matrix, then the equation Ax=b has at least one solution for each b in R^n

false. A needs to be invertible for this to be true

if A^T is not invertible, then A is not invertible

true

can a square matrix with two identical columns be invertible?

no; for 2×2 matrix and ab=ab→not invertible. nxn matrix, two identical columns→not linearly independent→not invertible

is it possible for a 5×5 matrix to be invertible when its columns do not span R^5

no; invertible←> spanning. not invertible←> not spanning

if the columns of a 5×5 matrix A are linearly independent, what can you say about the solution Ax=b?

yes; linearly independent 5×5 matrix→invertible→solution for each b for Ax=b

if the given equation Ax=y has more than one solution for some y in R^n, can the columns of A span R^n?

no; Ax=y has more than one solution→ Ax=0 has free variable, → cannot span R^n

if an 5×5 matrix A cannot be reduced to I5, what can you say about the columns of A?

doesnt span/not linearly independent

an nxn determinant is defined by determinants of (n-1) x (n-1) submatrices

true

the determinant of a triangular matrix is the sum of the entries of the main diagonal

false, not sum, should be multiplication

if det A is zero, then two rows or two columns are the same, or a row or a column is zero

false

if the columns of A are linearly dependent, then det A=0

true

the determinant of A is the product of the diagonal entries in A

false, A has to be triangular matrix

the determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U

false, scaling not taken into consideration

suppose A is an nxn matrix and a computer suggests that det A=5 and det (A^-1)= 1. should you trust these answers

false, det(A)= 5 then det(A^-1)= 1/5

suppose A is a 7x5 matrix, how many pivot columns must A have if its columns are linearly independent? why?

A matrix must have 5 pivot columns if its columns are linearly independent. Because has 5 columns, linear independence requires that every column is a pivot column (no free variables).

suppose A is a 7x5 matrix, how many pivot columns must A have if its columns span R^5? why?

A matrix must have 5 pivot columns if its columns span . Because the columns span a 5-dimensional space (), the rank of the matrix must be 5. Since a matrix cannot have more pivots than its number of columns, all 5 columns must be pivot columns.