Question
Show that if $A$ and $B$ are similar, then $\operatorname{det} A=\operatorname{det} B$
Step 1
Step 1: Since A and B are similar matrices, by definition, there exists an invertible matrix P such that $A = PBP^{-1}$. Show more…
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Key Concepts
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Let $A=\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$ and $B=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right] .$ Show that $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$ if and only if $a+d=0$
Determinants
Properties of Determinants
verify that $\operatorname{det}(A B)=\operatorname{det}(B A)$ and determine whether the equality det $(A+B)=\operatorname{det}(A)+\operatorname{det}(B)$ holds. $$A=\left[\begin{array}{lll} 2 & 1 & 0 \\ 3 & 4 & 0 \\ 0 & 0 & 2 \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ccc} 1 & -1 & 3 \\ 7 & 1 & 2 \\ 5 & 0 & 1 \end{array}\right]$$
Properties of Determinants; Cramer’s Rule
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