Linear Algebra, 2020a

Section III. Laplace’s Formula 367
The formulas from this subsection are often used for by-hand calculation
and are sometimes useful with special types of matrices. However, for generic
matrices they are not the best choice because they require more arithmetic than,
for instance, the Gauss-Jordan method.
Exercises
̌ 1.11 Find the cofactor.
⎛
1 0
⎞
2
T = ⎝−1 1 3 ⎠
0 2 −1
(a) T 2,3 (b) T 3,2 (c) T 1,3
̌ 1.12 Find the adjoint to this matrix.
⎛
⎞
1 0 2
T = ⎝−1 1 3 ⎠
0 2 −1
1.13 This determinant is 0. Compute that by expanding on the first row.
1 2 3
4 5 6
∣7 8 9∣
̌ 1.14 Find the determinant by expanding
3 0 1
1 2 2
∣−1 3 0∣
(a) on the first row (b) on the second row (c) on the third column.
1.15 Find the adjoint of the matrix in Example 1.6.
̌ 1.16 Find the matrix adjoint to each.
⎛ ⎞
⎛ ⎞
2 1 4 ( ) ( ) 1 4 3
(a) ⎝−1 0 2⎠
3 −1 1 1
(b)
(c)
(d) ⎝−1 0 3⎠
2 4
5 0
1 0 1
1 8 9
̌ 1.17 Find the inverse of each matrix in the prior question with Theorem 1.9.
1.18 Find the matrix adjoint to this one.
⎛
⎞
2 1 0 0
⎜1 2 1 0
⎟
⎝0 1 2 1⎠
0 0 1 2
̌ 1.19 Expand across the first row to derive the formula for the determinant of a 2×2
matrix.
̌ 1.20 Expand across the first row to derive the formula for the determinant of a 3×3
matrix.
̌ 1.21 (a) Give a formula for the adjoint of a 2×2 matrix.
(b) Use it to derive the formula for the inverse.
̌ 1.22 Can we compute a determinant by expanding down the diagonal?