Science:Math Exam Resources/Courses/MATH152/April 2012/Question 05 (b)

MATH152 April 2012

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[hide] Question 05 (b)
The following matrix A depends on a parameter p. Find all the values of of the parameter p for which the corresponding matrix is not invertible. $A={\begin{bmatrix}1&2&3&4\\100&200&300&401\\0&p&p+1&p+2\\0&-p&-2p+1&-3p+2\end{bmatrix}}$

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[show] Hint 1
The determinant is your friend. A matrix is noninvertible if its determinant is zero. For which values of p is the determinant of this matrix zero?

[show] Hint 2
To find the determinant simplify the matrix first by using row operations.

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[show] Solution
We begin by taking our matrix and applying row operations. First, we take 100 times the first row and subtract it from the second row to turn the matrix $A={\begin{bmatrix}1&2&3&4\\100&200&300&401\\0&p&p+1&p+2\\0&-p&-2p+1&-3p+2\end{bmatrix}}$ into ${\begin{bmatrix}1&2&3&4\\0&0&0&1\\0&p&p+1&p+2\\0&-p&-2p+1&-3p+2\end{bmatrix}}$ Next we subtract $p+2$ times the second row to the third row and also add $-3p+2$ times the second row to the fourth row giving ${\begin{bmatrix}1&2&3&4\\0&0&0&1\\0&p&p+1&0\\0&-p&-2p+1&0\end{bmatrix}}$ Next, we add the third row to the fourth row to get ${\begin{bmatrix}1&2&3&4\\0&0&0&1\\0&p&p+1&0\\0&0&-p+2&0\end{bmatrix}}$ Swap the second row with the third row and the resulting third row with the fourth row (so this introduces no change in the sign since we swap twice and so multiply the determinant $(-1)^{2}=1$ ) giving ${\begin{bmatrix}1&2&3&4\\0&p&p+1&0\\0&0&-p+2&0\\0&0&0&1\end{bmatrix}}$ Now we take determinants and notice that the determinant of A is the same as the determinant of the above matrix. Since the above matrix is upper triangular, the determinant can be taken by multiplying the entries on the diagonal to give $\det(A)=\det \left({\begin{bmatrix}1&2&3&4\\0&p&p+1&0\\0&0&-p+2&0\\0&0&0&1\end{bmatrix}}\right)=p(-p+2)$ This is zero when $\displaystyle p=0$ or when $\displaystyle p=2.$