Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 13

MATH152 April 2017

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[hide] Question A 13
Circle each of the following statements that is true for all invertible $n\times n$ matrices $A$ : (a) The rank of $A$ is $n$ . (b) $\det A=0$ . (c) The reduced row echelon form of $A$ is the $n\times n$ identity matrix. (d) $A^{-1}$ exists. (e) Columns of $A$ are linearly independent. (f) $A=A^{T}$ . .

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[show] Hint
Recall the definition of an invertible matrix and its properties.

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[show] Solution
Recall that we say a ${\textstyle n\times n}$ matrix invertible if ${\textstyle A^{-1}}$ exists. This implies that ${\textstyle \det(A)\neq 0}$ as ${\textstyle \det(A)\cdot \det(A^{-1})=\det(AA^{-1})=\det(I)=1.}$ Thus, ${\textstyle A}$ has full rank ${\textstyle n}$ , and the columns of ${\textstyle A}$ are linear independent. Note that any invertible matrix has the identity as its reduced row echelon form, thus $(c)$ is also correct. However, this does not necessarily imply that any invertible matrix is symmetric. Consider $A={\begin{bmatrix}2&2\\1&0\end{bmatrix}}.$ ${\textstyle A}$ is invertible but ${\textstyle A\neq A^{T}}$ .The answer is $\color {blue}{(a),(c),(d),(e)}$ .