Problem c let a beginbmatrix 3 4 0 0 0 1 0 0 2 2 1 0 2 2 2 3...

Problem C. Let $$ A = \begin{bmatrix} 3 & 4 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 2 & 3 \end{bmatrix} $$ Is $A^T$ similar to $A$? Justify your answer. Problem D. Let $A \in Mat_{n \times n}(F)$. Show that $$ dim_F Span_F \{A^k | k \in Z_{\geq 0}\} \leq n. $$ Here $A^0 = I_n$.

Transcript

00:01 The major theorem we're going to use is the invertible matrix theorem.

00:04 So i'll just use the abbreviation imt.

00:10 Okay, so the first statement, statement a, so we're given, so suppose there is an n -by -n matrix d such that a times d is the identity matrix, then there's also an m -by -n matrix c such that c times a is an identity.

00:29 So we're given then a times d is an identity.

00:36 So that means d is invertible.

00:46 Invertible.

00:50 And meanwhile, we know that a is exactly the inverse of d.

01:01 So that means a is also invertible.

01:17 So furthermore, since a is invertible, so there must exist.

01:22 By the imp, there must exist a matrix c, which is an unby -end matrix such that c c times a is identity.

01:46 So this is exactly what we are going to show, and that means this statement is true.

02:00 Okay, the next one.

02:05 Now for the statement b, if the columns of a are linearly independent, then the columns of a span rn.

02:17 So this is also true because this is just our imt, the invertible matrix function, the invertible matrix theorem.

02:31 We check our textbook, the statement e of imt is equivalent to statement h.

02:43 Of imt.

02:48 And that is exactly what was been said in statement b.

02:54 So that means this statement is true.

03:04 The next statement, c.

03:07 So if the equation, x equals b has at least one solution for each b, then the solution is unique for each b.

03:17 So we know that a x equals b has, as a at least one solution.

03:33 One solution.

03:35 So that means a be invertible.

03:49 A invertible...

Question

Problem C. Let $$ A = \begin{bmatrix} 3 & 4 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 2 & 2 & 1 & 0 \\ -2 & -2 & 2 & 3 \end{bmatrix} $$ Is $A^T$ similar to $A$? Justify your answer. Problem D. Let $A \in Mat_{n \times n}(F)$. Show that $$ dim_F Span_F \{A^k | k \in Z_{\geq 0}\} \leq n. $$ Here $A^0 = I_n$.

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