Matrix Algebra

Laplace Expansions

Laplace Expansions. The determinant of can be expanded in any of the following expansions,

$\begin{displaymath} \begin{array}{ll} \det A & = a_{i1} C_{i1} + a_{i2} C_{i2}... ...{nj} C_{nj} \quad\mbox{ for $j = 1,\ldots,n$. } \end{array} \end{displaymath}$

The expansions are often referred as Laplace expansions.

Example 2. Use the Laplace expansion across the third row to compute $\det {A}$ for $A = \begin{bmatrix} 1 & 5 & -2 \\ 2 & 4 &-1 \\ 0 &-2 & 0 \end{bmatrix}$

Determinant and Transpose Operation. The Laplace expansions with immediately indicate that the transpose has no effect in determinant, that is, that

$\displaystyle \det A^T = \det A.$

Determinant of Triangular Matrix. The determinant of a lower or upper triangular square matrix becomes the product of diagonal entries of the matrix. For example, the upper triangular square matrix has the determinant

$\displaystyle \det \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1... ... 0 & 0 & \cdots & a_{nn} \end{array}\right] = a_{11} a_{22} \cdots a_{nn}$