MATRIX Course

Leading minors of a matrix (example)

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This is an example of evaluating the leading minor of a matrix, as we explained on the previous page. Given the following matrix $A$,

\[ A = \begin{pmatrix} 4 & 2 & 2 \\ 2 & 10 & 7 \\ 2 & 7 & 21 \\ \end{pmatrix} \]

The leading minors of A are

  • $det(\Delta_1) = 4$
  • $det(\Delta_2) = 4 * 10 - 2 * 2 = 36$
  • $det(\Delta_3)$
    • $= 4 * (10 * 21 - 7 * 7) - 2 * (2* 21 -2 * 7) + 2 * ( 2 * 7 - 2 * 10)$
    • $= 4 * 161 - 68 = 3 * 161 + 93$
    • $= (3*16)*10 + 100 - 7 + 3 = 576$

Every leading minor is greater than $0$, so the matrix is positive definite.