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

MATH152 April 2013

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[hide] 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.

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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!

[show] 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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[show] Solution

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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