Answer to Question #245553 in Linear Algebra for moe

Question #245553

You are given that A and B are two square matrices of the same order, such that


"\\mathrm{det}\\left(A^{-1}B\\right)=20\\quad \\mathrm{and} \\quad \\mathrm{det} \\left(B\\right)=5."


Which of the following is true?

  1. "\\mathrm{det}(A)=-15\\quad \\text{and} \\quad \\mathrm{adj}(A)=-\\frac{1}{15}"
  2. "\\mathrm{det}(A)=4\\quad \\text{and} \\quad\\mathrm{adj}(A)=4A^{-1}"
  3. "\\mathrm{det}(A)=\\frac{1}{4}\\quad \\text{and} \\quad \\mathrm{adj}(A)=\\frac{1}{4}A^{-1}"
  4. "\\mathrm{det}(A)=25\\quad \\text{and} \\quad \\mathrm{adj}(A)=25"
  5. "\\mathrm{det}(A)=15\\quad \\text{and} \\quad \\mathrm{adj}(A)=15 A^{-1}"

.


1
Expert's answer
2021-10-06T16:20:55-0400

"\\det(A^{-1}B)=\\det{A^{-1}}\\det{B}\\\\=\\dfrac{\\det{B}}{\\det{A}}\\\\\n20=\\dfrac{5}{\\det{A}}\\\\\n\\det{A}=\\dfrac{1}{4}\\\\\n\\text{Inverse formula }A^{-1}=\\dfrac{\\text{adj(A)}}{\\det(A)}\\\\\n\\text{adj(A)}=(\\det{A})A^{-1}=\\dfrac{1}{4}A^{-1}"

True statement is 3


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