Exercise · ZW-COFACTOR-LAPLACE-EXPANSION-010 · ★★★ Advanced
Find the values of \(x\) for which \(\det\begin{pmatrix}x & 1 & 0 \\ 2 & x & 1 \\ 0 & 1 & x\end{pmatrix}=0\).
Choose one option.
Show answer
- C. \(x=0,\ x=\pm\sqrt3\)
Explanation
Along the first row: \(x(x^2-1)-1\cdot(2x-0)=x^3-3x=x(x^2-3)\), which vanishes at \(x=0\) and \(x=\pm\sqrt3\).
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…