Answer:
k = -6
w = ±9
f = -7
Determinant = -136
Step-by-step explanation:
[tex]\left[\begin{array}{cc}-12&-w^{2}\\2f&3\end{array}\right] =\left[\begin{array}{cc}2k&-81\\-14&3\end{array}\right][/tex]
Each cell in the left matrix equals the corresponding cell in the right matrix:
-12 = 2k → k = -6
-w² = -81 → w = ±9
2f = -14 → f = -7
[tex]\left|\begin{array}{ccc}-4&5&6\\0&4&4\\-2&-5&4\end{array}\right|[/tex]
To find the determinant of a 3x3 matrix, you can use something called "Laplace expansion".
Start with the first column in the top row (-4). If you block out the row and column containing that cell, you get a 2x2 matrix:
[tex]\left|\begin{array}{ccc}*&*&*\\ *&4&4\\ *&-5&4\end{array}\right|[/tex]
Multiply the -4 by the determinant of that 2x2 matrix:
[tex]-4\left|\begin{array}{cc}4&4\\-5&4\end{array}\right|[/tex]
Repeat for the other two cells in the top row.
[tex]5\left|\begin{array}{cc}0&4\\-2&4\end{array}\right|[/tex]
[tex]6\left|\begin{array}{cc}0&4\\-2&-5\end{array}\right|[/tex]
Add them together, alternating the signs (first column positive, second column negative, third column positive).
[tex]-4\left|\begin{array}{cc}4&4\\-5&4\end{array}\right|-5\left|\begin{array}{cc}0&4\\-2&4\end{array}\right|+6\left|\begin{array}{cc}0&4\\-2&-5\end{array}\right|[/tex]
To find the determinants of the 2x2 matrices, multiply the top left and bottom right, then subtract the top right times the bottom left.[tex]-4((4\times4)-(4\times-5))-5((0\times4)-(4\times-2))+6((0\times-5)-(4\times-2))[/tex]
Simplify:
[tex]-4(16-(-20))-5(0-(-8))+6(0-(-8))\\-4(36)-5(8)+6(8)\\-144-40+48\\-136[/tex]
