Laplace’s Expansion

Find the cofactor.

• (a)  $T_{2,3}$     (b)  $T_{3,2}$     (c)  $T_{1,3}$

Find the adjoint to this matrix.

This determinant is $0$. Compute that by expanding on the first row.

Find the determinant by expanding

• (a)  on the first row     (b)  on the second row     (c)  on the third column.

Find the adjoint of the matrix in Example 1.5.

Find the matrix adjoint to each.

• (a)  $\begin{pmatrix}2&1&4\\ -1&0&2\\ 1&0&1\end{pmatrix}$     (b)  $\begin{pmatrix}3&-1\\ 2&4\end{pmatrix}$     (c)  $\begin{pmatrix}1&1\\ 5&0\end{pmatrix}$     (d)  $\begin{pmatrix}1&4&3\\ -1&0&3\\ 1&8&9\end{pmatrix}$

Find the inverse of each matrix in the prior question with Theorem 1.8.

Find the matrix adjoint to this one.

Expand across the first row to derive the formula for the determinant of a $2\!\times\!2$ matrix.

Expand across the first row to derive the formula for the determinant of a $3\!\times\!3$ matrix.

• Give a formula for the adjoint of a $2\!\times\!2$ matrix.

• Use it to derive the formula for the inverse.

Can we compute a determinant by expanding down the diagonal?

Give a formula for the adjoint of a diagonal matrix.

Prove that the transpose of the adjoint is the adjoint of the transpose.

Prove or disprove: $\operatorname{adj}(\operatorname{adj}(T))=T$.

A square matrix is upper triangular if each $i,j$ entry is zero in the part above the diagonal, that is, when $i>j$.

• Must the adjoint of an upper triangular matrix be upper triangular? Lower triangular?

• Prove that the inverse of a upper triangular matrix is upper triangular, if an inverse exists.

This question requires material from the optional Determinants Exist subsection. Prove Theorem 1.4 by using the permutation expansion.

Prove that the determinant of a matrix equals the determinant of its transpose using Laplace’s expansion and induction on the size of the matrix.

Show that

where $F_{n}$ is the $n$-th term of $1,1,2,3,5,\dots,x,y,x+y,\ldots\,$, the Fibonacci sequence, and the determinant is of order $n-1$.