Shilov's Linear Algebra - Chapter 1, Problem 9

Question

Calculate the $n$-th order determinant:

$$\Delta= \begin{vmatrix} x&a&a&\ldots&a\\ a&x&a&\ldots&a\\ a&a&x&\ldots&a\\ \cdot&\cdot&\cdot&\ldots&\cdot\\ a&a&a&\ldots&x \end{vmatrix}$$

The answer is $\Delta=[x+a(n-1)](x-a)^{n-1}$.

If we add all the other columns to the first column, we get the first multiplicative factor of the answer, and are left with the following determinant:

$$\begin{vmatrix} 1&a&a&\ldots&a\\ 1&x&a&\ldots&a\\ 1&a&x&\ldots&a\\ \cdot&\cdot&\cdot&\ldots&\cdot\\ 1&a&a&\ldots&x \end{vmatrix}$$

How can we calculate this determinant to obtain the answer?

Hint: add the first column multiplied by $-a$ to the other columns. — Timur Bakiev, Jun 10 '21 at 07:04
Subtract the last row from all other rows. Then to the last column add all other columns. — Joe Mansley, Jun 10 '21 at 08:53

C Squared · Answer 1 · 2021-06-10T08:03:20.167

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If the matrix $J$ is the square $n\times n$ matrix with all $1$'s as its entries, then your matrix, call it $M$, is given by $M=(x-a)I+aJ$.

The eigen values of $J$ are $n$ once and $0$, $n-1$ times, so the eigenvalues of $aJ$ are $an$ once and $0$, $n-1$ times. Therefore the eigenvalues of $(x-a)I+aJ$ are $an+x-a$ once and $x-a$, $n-1$ times.

The determinant of $M$ is the product of the eigenvalues of $M$, hence $$\Delta=[x+a(n-1)](x-a)^{n-1}$$

edited Jun 10 '21 at 08:03

answered Jun 10 '21 at 07:16

C Squared

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This question is asked every 3 months or so (see the reference I gave). – Jean Marie Jun 10 '21 at 08:44
@JeanMarie are suggesting I delete my answer or...? – C Squared Jun 10 '21 at 08:45
No don't delete it ! It was just a remark. – Jean Marie Jun 10 '21 at 08:46
@JeanMarie ok, gotcha. it’s hard to pick up tone of voice over the internet sometimes – C Squared Jun 10 '21 at 08:49
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It's right that interpretation of intentions is eased when you hear the voice or, even better, see the face of people... For a future release of Maths SE... – Jean Marie Jun 10 '21 at 08:52

Shilov's Linear Algebra - Chapter 1, Problem 9

1 Answers1