Exercise · ZW-COFACTOR-LAPLACE-EXPANSION-009 · ★★★ Advanced

To compute the determinant of a \(4\times4\) matrix by expanding along a row with no zeros, how many \(3\times3\) determinants (minors) must be evaluated?

Choose one option.

Show answer

B. \(4\)

Explanation

Each entry of the row contributes a term \(a_{ij}C_{ij}\) with one \(3\times3\) minor: 4 entries, so 4 minors. Each zero would save one.

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Review the concept: Cofactor and Laplace expansion

The minor \(M\) of an entry is the determinant of the submatrix left after deleting its row and column; the cofactor \(C\) is that minor with an alternating sign.

\[C_{ij}=(-1)^{i+j}M_{ij}\]

More practice problems: cofactor and laplace expansion

ZW-COFACTOR-LAPLACE-EXPANSION-001 For A=[1 2 3; 4 5 6; 7 8 10], compute the cofactor C_12.
ZW-COFACTOR-LAPLACE-EXPANSION-002 For the same matrix A=[1 2 3; 4 5 6; 7 8 10], compute the minor M_23 (no sign).
ZW-COFACTOR-LAPLACE-EXPANSION-003 Which sign (-1)^i+j corresponds to position (3,2)?
ZW-COFACTOR-LAPLACE-EXPANSION-004 Compute det B by expanding along the first row, with B=[2 0 1; 3 1 0; 1 2 4].
ZW-COFACTOR-LAPLACE-EXPANSION-005 Compute the cofactor C_21 of [2 3; 5 7].
ZW-COFACTOR-LAPLACE-EXPANSION-006 Compute det[1 0 2; 3 0 1; 4 5 6] by expanding along the most convenient column.
ZW-COFACTOR-LAPLACE-EXPANSION-007 Compute the determinant of [2 0 0 0; 1 3 0 0; 4 5 6 0; 7 8 9 1].
ZW-COFACTOR-LAPLACE-EXPANSION-008 For A=[1 2 3; 4 5 6; 7 8 10], what is a_11C_21+a_12C_22+a_13C_23 (row 1 with the cofactors of row 2…