Science:Math Exam Resources/Courses/MATH152/April 2013/Question A 12

MATH152 April 2013

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Question A 12
Let $A={\begin{bmatrix}1&3\\3&8\end{bmatrix}}\qquad B={\begin{bmatrix}1&0&-1\\-3&1&0\end{bmatrix}}$ Calculate the following whenever defined. Otherwise state undefined and include a brief explanation. A matrix $D$ such that $DB=I_{3}$ where $I_{3}$ is the $3\times 3$ identity matrix.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Proceed similarly to question A11. Start off with a general matrix $D={\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}$ and perform the multiplication.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We proceed as in the hint. Let $D={\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}$ Then we have ${\begin{aligned}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=DB&={\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}}{\begin{bmatrix}1&0&-1\\-3&1&0\end{bmatrix}}\\&={\begin{bmatrix}(a)(1)+(b)(-3)&(a)(0)+(b)(1)&(a)(-1)+(b)(0)\\(c)(1)+(d)(-3)&(c)(0)+(d)(1)&(c)(-1)+(d)(0)\\(e)(1)+(f)(-3)&(e)(0)+(f)(1)&(e)(-1)+(f)(0)\end{bmatrix}}\\&={\begin{bmatrix}a-3b&b&-a\\c-3d&d&-c\\e-3f&f&-e\end{bmatrix}}\end{aligned}}$ Looking at the first row on the matrices on the left and right gives ${\begin{aligned}a-3b&=1\\b&=0\\-a&=0\end{aligned}}$ and this is inconsistent since the last two imply that both a and b are zero and this contradicts the first line. Thus no such matrix D exists.

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