Examples on Determinants and Inverses

Comp.145  November 1996  Dr. A.J. Mestel

Assessed work: Hand in solutions to the questions marked with a ^{∗} to the general office by Wednesday 6th November.

∗1. Determine whether or not the three vectors (1, 1, −1)^{T}, (1, −2, 1)^{T} and (3, 0, −1)^{T} are linearly dependent.

∗2. Find the cofactors and hence the inverse of the matrix ( \begin{matrix}2 & 0 & 3 \\ 4 & 1 & 1 \\ 2 & 1 & −1 \end{matrix} ). Use the

inverse matrix to solve the equations

\begin{matrix}2x + 3z & = −1 \\ 4x + y + z & = 1 \\ 2x + y − z & = 1 \end{matrix} \} .

(Include your working.)

∗3. If A, B, C are square matrices such that BA = I and AC = I, prove that B = C. [N.B. Do not assume any properties of A^{−1}. You need only associativity and properties of I.]

∗4. A recursive algorithm for evaluating an n × n determinant requires calculating n determinants of size (n − 1) × (n − 1). Thus, if T_{n} is the number of multiplications required, then approximately T_{n} = nT_{n−1} for n ⩾ 3. Estimate how long it would take a Cray, capable of 10^{9} such operations per second, to evaluate a 25 × 25 determinant using this method.

5. The n × n tri-diagonal matrix A_{n} takes the form

A_{n} = ( \begin{matrix}a & b & 0 & 0 & ... & 0 & 0 \\ c & a & b & 0 & ... & 0 & 0 \\ 0 & c & a & b & ... & 0 & 0 \\ 0 & 0 & c & a & ... & 0 & 0 \\ ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & 0 & ... & a & b \\ 0 & 0 & 0 & 0 & ... & c & a \end{matrix} )  ↑ n rows.

If D_{n} = detA_{n}, show that D_{n} = aD_{n−1} − bcD_{n−2}.

Suppose now that a = 3, b = 1 and c = 2. Later in the course you will be able to show that for every n, D_{n} = B2^{n} + C, where B and C are suitable constants (they do not vary with n). Assuming this is the case, find B and C.