Logarithm 11

Calculus Level 3

$\begin{aligned} f(x,y,z) & = \frac{\ln(y^x)}{z} \\ g(x,y) & = e^{- x^2 - y^2} \\ A & = \begin{bmatrix} g(0,1) & f(0,3,5) & g(0,-1) \\ f(4,e,2) & g(0,0) & f(3,e^3,3) \\ g(1,0) & f(7,1,1) & g(-1,0) \\ \end{bmatrix} \end{aligned}$

Given the functions $f$ and $g$ and matrix $A$ above, find the determinant of $A$ .

-1

e

1 0

\frac{1}{e}

1 solution

Tom Engelsman
May 17, 2017

Computing the appropriate values of the matrix A yields:

$\begin{aligned} A & = \begin{bmatrix} e^{-1} & 0 & e^{-1} \\ 2 & 1 & 3 \\ e^{-1} & 0 & e^{-1} \\ \end{bmatrix} \end{aligned}$

Upon observation, the third column vector equals the sum of the first and the second column vectors (i.e. a linear combination). Since A does not have full rank, its determinant is singular $\Rightarrow det(A) = \boxed{0}.$

Logarithm 11

1 solution

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