Find the determinant of [[1,4,-3],[-4,-8,-4],[-7,6,3]]: animated lesson

1. 1. The problem
   Worked example

   $\det \begin{bmatrix}1 & 4 & -3 \\ -4 & -8 & -4 \\ -7 & 6 & 3\end{bmatrix}$
2. 2. Step by step
   1. $$A = \begin{bmatrix}1 & 4 & -3 \\ -4 & -8 & -4 \\ -7 & 6 & 3\end{bmatrix}$$
      Read the 3×3 matrix
   2. $$\det A = (1)\begin{vmatrix}-8 & -4 \\ 6 & 3\end{vmatrix} - (4)\begin{vmatrix}-4 & -4 \\ -7 & 3\end{vmatrix} + (-3)\begin{vmatrix}-4 & -8 \\ -7 & 6\end{vmatrix}$$
      Expand along the first row (cofactor expansion)
   3. $$M_{11} = 0,\quad M_{12} = -40,\quad M_{13} = -80$$
      Evaluate each 2×2 minor
   4. $$\det A = (1)(0) - (4)(-40) + (-3)(-80) = 400$$
      Combine with the alternating signs + − +
3. 3. The answer
   $$\det A = 400$$

   det = 400

4. 4. Check
   • ✓Computed exactly with fractions; row reduction gives the same value 400.

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