Section 12

Expanding a determinant according to the elements of a row or a column

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Problem 12.1

Expanding along the third row, compute the determinant

[2−3414−232abcd3−143] \left[\begin{smallmatrix} 2 & -3 & 4 & 1 \\ 4 & -2 & 3 & 2 \\ a & b & c & d \\ 3 & -1 & 4 & 3 \end{smallmatrix}\right]
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Problem 12.2

Expanding along the second column, compute the determinant

[5a2−14b4−32c3−24d5−4] \left[\begin{smallmatrix} 5 & a & 2 & -1 \\ 4 & b & 4 & -3 \\ 2 & c & 3 & -2 \\ 4 & d & 5 & -4 \end{smallmatrix}\right]
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Problem 12.3

Compute the determinants:

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(a)

[xy0…000xy…0000x…00………………000…xyy00…0x]\left[\begin{smallmatrix} x & y & 0 & \ldots & 0 & 0 \\ 0 & x & y & \ldots & 0 & 0 \\ 0 & 0 & x & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & x & y \\ y & 0 & 0 & \ldots & 0 & x\end{smallmatrix}\right];

(b)

[a0a1a2…an−1an−y1x10…000−y2x2…00…………………000…xn−10000…−ynxn]\left[\begin{smallmatrix} a_{0} & a_{1} & a_{2} & \ldots & a_{n-1} & a_{n} \\ -y_{1} & x_{1} & 0 & \ldots & 0 & 0 \\ 0 & -y_{2} & x_{2} & \ldots & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & x_{n-1} & 0 \\ 0 & 0 & 0 & \ldots & -y_{n} & x_{n}\end{smallmatrix}\right];

(c)

[a0−100…00a1x−10…00a20x−1…00……………………….an−1000…x−1an000…0x]\left[\begin{smallmatrix} a_{0} & -1 & 0 & 0 & \ldots & 0 & 0 \\ a_{1} & x & -1 & 0 & \ldots & 0 & 0 \\ a_{2} & 0 & x & -1 & \ldots & 0 & 0 \\ \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots & \ldots & \ldots & . \\ a_{n-1} & 0 & 0 & 0 & \ldots & x & -1 \\ a_{n} & 0 & 0 & 0 & \ldots & 0 & x\end{smallmatrix}\right];

(d)

[n!a0(n−1)!a1(n−2)!a2…an−nx0…an0−(n−1)x…0…………………………………]\left[\begin{smallmatrix} n!a_{0} & (n-1)!a_{1} & (n-2)!a_{2} & \ldots & a_{n} \\ -n & x & 0 & \ldots & a_{n} \\ 0 & -(n-1) & x & \ldots & 0 \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots & \ldots & \ldots \end{smallmatrix}\right];

(e)

[123…n−1n−1x0…000−1x…00…………………000…x0000…−1x]\left[\begin{smallmatrix} 1 & 2 & 3 & \ldots & n-1 & n \\ -1 & x & 0 & \ldots & 0 & 0 \\ 0 & -1 & x & \ldots & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & x & 0 \\ 0 & 0 & 0 & \ldots & -1 & x\end{smallmatrix}\right];

(f)

[n−100…00n−1x−10…00n−20x−1…00……………………2000…x−11000…0x]\left[\begin{smallmatrix} n & -1 & 0 & 0 & \ldots & 0 & 0 \\ n-1 & x & -1 & 0 & \ldots & 0 & 0 \\ n-2 & 0 & x & -1 & \ldots & 0 & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 2 & 0 & 0 & 0 & \ldots & x & -1 \\ 1 & 0 & 0 & 0 & \ldots & 0 & x\end{smallmatrix}\right];

(g)

[1000…011a100…0011a20…00101a3…00…………………1000…1an]\left[\begin{smallmatrix} 1 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 1 & a_{1} & 0 & 0 & \ldots & 0 & 0 \\ 1 & 1 & a_{2} & 0 & \ldots & 0 & 0 \\ 1 & 0 & 1 & a_{3} & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 0 & 0 & 0 & \ldots & 1 & a_{n}\end{smallmatrix}\right];

(h)

[a10…0b10a2…b20………………]\left[\begin{smallmatrix} a_{1} & 0 & \ldots & 0 & b_{1} \\ 0 & a_{2} & \ldots & b_{2} & 0 \\ \ldots \ldots & \ldots & \ldots & \ldots & \ldots \end{smallmatrix}\right];

(i)

[a0111…11a100…010a20…0………………1000…an]\left[\begin{smallmatrix} a_{0} & 1 & 1 & 1 & \ldots & 1 \\ 1 & a_{1} & 0 & 0 & \ldots & 0 \\ 1 & 0 & a_{2} & 0 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 0 & 0 & 0 & \ldots & a_{n}\end{smallmatrix}\right];

Problem 12.4

Prove that the (n+1)(n+1)-st term u(n+1)u(n+1) of the sequence of Fibonacci numbers (see 4.5) equals the determinant

[1100…00−1110…000−111…00…………………0000…−11] \left[\begin{smallmatrix} 1 & 1 & 0 & 0 & \ldots & 0 & 0 \\ -1 & 1 & 1 & 0 & \ldots & 0 & 0 \\ 0 & -1 & 1 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & -1 & 1 \end{smallmatrix}\right]

of order nn.

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