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The problem provides determinants for two 3x3 matrices and asks for the determinants of two additional matrices. Let's go through this step-by-step.

### Given Information
1. \(\det \begin{bmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \end{bmatrix} = -5\)
2. \(\det \begin{bmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{bmatrix} = 4\)

### Step 1: First Determinant
To find the determinant of \(\begin{bmatrix} a & 9 & d \\ b & 9 & e \\ c & 9 & f \end{bmatrix}\):

Observe that each element in the second column is now 9. Since adding a multiple of one row to another row does not change the determinant, we can rewrite this determinant by factoring out 9 from the second column. This effectively adds 8 times the original matrix to the second column without affecting the determinant calculation due to linearity of determinant calculations.



If det[[a, 1, d], [b, 1, e], [c, 1, f]] = -5 and det[[a, 1, d], [b, 2, e], [c, 3, f]] = 4, then find det[[a, 9, d], [b, 9, e], [c, 9, f]] and det[[a, 0, d], [b, -3, e], [c, -6, f]].

Learn how to find the determinants of 3x3 matrices with specific values. This problem involves using determinant properties and matrix operations, ideal for Grade 12 or college-level linear algebra students.

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