c5a423…: “Near the fall chord”

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The Invertible Matrix Theorem: Let $A$ be a square $n \times n$ matrix. Then the following statements are equivalent.

(a): $A$ is an invertible matrix.
(b): $A$ is row equivalent to the $n \times n$ identity matrix.
(c): $A$ has $n$ pivot positions.
(d): The equation $A\vec {x} = \vec {0}$ has only the trivial solution.
(e): The columns of $A$ form a linearly independent set.
(f): The linear transformation $\vec {x} \mapsto A\vec {x}$ is one-to-one.
(g): The equation $A\vec {x} = \vec {b}$ has at least one solution for each $\vec {b}$ in $\RR ^n$ .
(h): The columns of $A$ span $\RR ^n$ .
(i): The linear transformation $\vec {x} \mapsto A\vec {x}$ maps $\RR ^n$ onto $\RR ^n$ .
(j): There is an $n \times n$ matrix $C$ such that $CA =I$ .
(k): There is an $n\times n$ matrix $D$ such that $AD = I$ .
(l): $A^T$ is an invertible matrix.

Determine if the matrices below are invertible.

A square matrix that is row equivalent to $I_8$ .

Invertible Not Invertible

$A = \begin {bmatrix} 1& 2& 0 \\ 0& 3& -1\\ 0&0&4 \end {bmatrix}$ .

Invertible Not Invertible

$A = \begin {bmatrix} 1& 8& 0 &5 \\ 0& 6& 0&3\\ 0&0&1& 1 \end {bmatrix}$ .

Invertible Not Invertible

$A = \begin {bmatrix} 1& 8& 0 &5 \\ 0& 6& 0&3\\ 0&0&1& 1 \\ 0&0&0&0 \end {bmatrix}$ .

Invertible Not Invertible

$A = \begin {bmatrix} 5& 0& 0 \\ -3& -7& 0\\ 8&5&-1 \end {bmatrix}$ . (Hint: $A$ is not in echelon form)

Invertible Not Invertible