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Clarence
 one year ago
Suppose A is an invertible matrix. Which of the following statements are true? Explanations would be greatly appreciated!
(a) A can be expressed as a product of elementary matrices.
(b) A is row equivalent to the nxn identity matrix.
(c) The equation Ax=0 has only the trivial solution.
(d) The determinant of A is positive.
(e) The transpose of A is also an invertible matrix.
(f) A has a unique row echelon form.
(g) The equation Ax=b may have two distinct non zero solutions for a non zero vector b.
Clarence
 one year ago
Suppose A is an invertible matrix. Which of the following statements are true? Explanations would be greatly appreciated! (a) A can be expressed as a product of elementary matrices. (b) A is row equivalent to the nxn identity matrix. (c) The equation Ax=0 has only the trivial solution. (d) The determinant of A is positive. (e) The transpose of A is also an invertible matrix. (f) A has a unique row echelon form. (g) The equation Ax=b may have two distinct non zero solutions for a non zero vector b.

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0a, b, c and e are true. d and g are not true f i'm still debating

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i think f is false. if it were rowreduced echelon form then it would be true. Suppose\[A \rightarrow \left[\begin{matrix}1 & 2 \\ 0 & 1\end{matrix}\right] \rightarrow \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\] both are in row echelon form (the last is in rowreduced echelon form) thus row echelon form is not unique

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Are you familiar with this property of determinants: \[det(AB)=det(A)*det(B)\] So if a matrix is invertible, then: \(AA^{1} = I\) \[det(AA^{1}) = det(I)\] \[det(A)*det(A^{1}) = 1\] So we can rearrange this, \[det(A^{1}) = \frac{1}{det(A)}\] So if the determinant of A is 0, then it doesn't have an inverse, because you can't divide by zero to find the determinant of of its inverse. Also you get this fancy little relationship between determinants.
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