Math Problems Vault

Problem 2588

Evaluate the determinant

\begin{vmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{vmatrix} .

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Problem 2589

Evaluate the determinant

\begin{vmatrix} 10 & -5 & 10 \\ 11 & -2 & 10 \\ 2 & -14 & -5 \end{vmatrix} .

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Problem 2590

Evaluate the determinant

\begin{vmatrix} 3 & 6 & 2 \\ 2 & 3 & 6 \\ 6 & 2 & 3 \end{vmatrix} .

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Problem 2591

Evaluate the determinant

\begin{vmatrix} 5 & 0 & 4 & 2 \\ 1 & -1 & 2 & 1 \\ 4 & 1 & 2 & 0 \\ 1 & 1 & 1 & 1 \end{vmatrix} .

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Problem 2592

Evaluate the determinant

\begin{vmatrix} 3 & 1 & 2 & 3 \\ 4 & -1 & 2 & 4 \\ 1 & -1 & 1 & 1 \\ 4 & -1 & 2 & 5 \end{vmatrix} .

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Problem 2593

Evaluate the determinant

\begin{vmatrix} 1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49 \end{vmatrix} .

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Problem 2594

Evaluate the determinant

\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+a & 1 \\ 1 & 1 & 1+b \end{vmatrix} .

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Problem 2595

Evaluate the determinant

\begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix} .

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Problem 2596

Evaluate the determinant

\begin{vmatrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{vmatrix} .

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Problem 2597

Evaluate the determinant

\begin{vmatrix} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{vmatrix} .

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Problem 2598

Evaluate the determinant

\begin{vmatrix} 1 & c & -b \\ -c & 1 & a \\ b & -a & 1 \end{vmatrix} .

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Problem 2599

Evaluate the determinant

\begin{vmatrix} a & 1 & 0 & 0 \\ -1 & b & 1 & 0 \\ 0 & -1 & c & 1 \\ 0 & 0 & -1 & d \end{vmatrix} .

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Problem 2600

Evaluate the determinant

\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^{2} & y^{2} & z^{2} \end{vmatrix} .

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Problem 2601

Find the value of the Vandermonde determinant

\Delta =\begin{vmatrix} 1 & 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & x_{3} & \ldots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ x_{1}^{n-1} & x_{2}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix},

assuming first that the numbers $x_{1}, x_{2}, \ldots , x_{n}$ are pairwise distinct.

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Problem 2602

Evaluate the determinants $\Delta_{n}$ and $\Delta_{n+1}$ obtained from the Vandermonde determinant (see problem 2601) by replacing, in the last row, the exponent $n-1$ by $n$ or by $n+1$ .

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Problem 2603

Evaluate the values $\Delta_{1}$ and $\Delta_{2}$ of the determinants obtained from the Vandermonde determinant $\Delta$ (see problem 2601) by differentiating with respect to $x_{1}$ and replacing $x_{1}$ by $x_{2}$ , or $x_{1}$ and $x_{2}$ by $x_{3}$ :

\Delta _{1}=\begin{vmatrix} 0 & 1 & 1 & \ldots & 1 \\ 1 & x_{2} & x_{3} & \ldots & x_{n} \\ 2x_{2}^{2} & x_{2}^{2} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ (n-1)x_{2}^{n-1} & x_{2}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix};

\Delta _{2}=\begin{vmatrix} 0 & 0 & 1 & \ldots & 1 \\ 0 & 1 & x_{3} & \ldots & x_{n} \\ 2 & 2x_{3} & x_{3}^{2} & \ldots & x_{n}^{2} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ n(n-1)x_{3}^{n-2} & (n-1)x_{3}^{n-1} & x_{3}^{n-1} & \ldots & x_{n}^{n-1} \end{vmatrix} .

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Problem 2604

Prove the equality:

\begin{vmatrix} 1 & 1 & \ldots & 1 \\ \binom {x_{1}}{1} & \binom {x_{2}}{1} & \ldots & \binom {x_{n}}{1} \\ \binom {x_{1}}{2} & \binom {x_{2}}{2} & \ldots & \binom {x_{n}}{2} \\ \vdots & \vdots & \vdots & \vdots \\ \binom {x_{1}}{n-1} & \binom {x_{2}}{n-1} & \ldots & \binom {x_{n}}{n-1} \end{vmatrix}=\dfrac {\Delta }{1!2!3!\ldots (n-1)!},

where $\dbinom {x}{m}=\dfrac {x(x-1)\ldots (x-m+1)}{1 \cdot 2 \ldots m}$ , $\Delta$ is the Vandermonde determinant (see problem 2601).

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Problem 2605

Evaluate the determinant

\begin{vmatrix} \cos (a-b) & \cos (b-c) & \cos (c-a) \\ \cos (a+b) & \cos (b+c) & \cos (c+a) \\ \sin (a+b) & \sin (b+c) & \sin (c+a) \end{vmatrix} .

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Problem 2606

Evaluate the determinant

\begin{vmatrix} \sin a & \cos a & \sin 2a \\ \sin b & \cos b & \sin 2b \\ \sin c & \cos c & \sin 2c \end{vmatrix} .

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Problem 2607

Evaluate the determinant

\begin{vmatrix} \sin a & \sin b & \sin c \\ \sin 3a & \sin 3b & \sin 3c \\ \sin 5a & \sin 5b & \sin 5c \end{vmatrix} .

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Problem 2608

Evaluate the determinant

\begin{vmatrix} \sin ^{3}\alpha & \sin ^{2}\alpha \cos \alpha & \sin \alpha \cos ^{2}\alpha & \cos ^{3}\alpha \\ \sin ^{3}\beta & \sin ^{2}\beta \cos \beta & \sin \beta \cos ^{2}\beta & \cos ^{3}\beta \\ \sin ^{3}\gamma & \sin ^{2}\gamma \cos \gamma & \sin \gamma \cos ^{2}\gamma & \cos ^{3}\gamma \\ \sin ^{3}\delta & \sin ^{2}\delta \cos \delta & \sin \delta \cos ^{2}\delta & \cos ^{3}\delta \end{vmatrix} .

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Problem 2609

Prove the equality: $D_{n}=a_{n}D_{n-1}+D_{n-2}$ ;

D_{n}=\begin{vmatrix} a_{1} & 1 & 0 & \ldots & 0 & 0 \\ -1 & a_{2} & 1 & \ldots & 0 & 0 \\ 0 & -1 & a_{3} & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & a_{n-1} & 1 \\ 0 & 0 & 0 & \ldots & -1 & a_{n} \end{vmatrix} .

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Problem 2610

Prove the equality

\begin{vmatrix} a_{1} & \lambda & \ldots & \lambda \\ \lambda & a_{2} & \ldots & \lambda \\ \vdots & \vdots & \vdots & \vdots \\ \lambda & \lambda & \ldots & a_{n} \end{vmatrix}=\varphi (\lambda )-\lambda \dfrac {d\varphi }{d\lambda },

where $\varphi (\lambda )=(a_{1}-\lambda )(a_{2}-\lambda )\ldots (a_{n}-\lambda )$ .

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Problem 2611

Prove the equality

\begin{vmatrix} a_{1} & a_{2} & a_{3} & \ldots & a_{n} \\ a_{2} & a_{3} & a_{4} & \ldots & a_{1} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n} & a_{1} & a_{2} & \ldots & a_{n-1} \end{vmatrix}=(-1)^{\frac{(n-1)(n-2)}{2}}\varphi (\alpha _{1})\varphi (\alpha _{2})\ldots \varphi (\alpha _{n}),

where $\varphi (\alpha )=a_{1}+a_{2}\alpha +\ldots +a_{n}\alpha^{n-1}$ ; $\alpha$ is a root of the equation $\alpha^{n}=1$ .

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Problem 2612

Prove the equality

\begin{vmatrix} 1 & 2 & 3 & \ldots & n \\ 2 & 3 & 4 & \ldots & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ n & 1 & 2 & \ldots & n-1 \end{vmatrix}=(-1)^{\frac{n(n-1)}{2}} \cdot \dfrac {n(n+1)}{2}n^{n-2} .

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Problem 2613

Prove the inequality:

\Delta =\begin{vmatrix} A_{1} & a_{2} & a_{3} & \ldots & a_{n} \\ b_{1} & B_{2} & b_{3} & \ldots & b_{n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ m_{1} & m_{2} & m_{3} & \ldots & M_{n} \end{vmatrix} \neq 0,

where $A_{1} > \left|a_{2}\right|+\left|a_{3}\right|+\ldots +\left|a_{n}\right|$ , $B_{2} > \left|b_{1}\right|+\left|b_{3}\right|+\ldots +\left|b_{n}\right|, \ldots$ , $M_{n} > \left|m_{1}\right|+\left|m_{2}\right|+\ldots +\left|m_{n-1}\right|$ .

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Problem 2614

Prove that the determinant of the preceding problem (see problem 2613) is positive, if, under the same conditions, its entries are real.

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Problem 2615

Prove that a determinant whose entries $a_{\nu \mu }$ and $a_{\mu \nu }$ , symmetric with respect to the main diagonal, are complex conjugates, has a real value.

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Problem 2616

The entries of a determinant of order $n$ are related by $a_{\nu \nu }=0$ , $a_{\nu \mu }=ia_{\mu \nu }$ , $\nu > \mu$ , where $a_{\nu \mu }$ is a real number. Find the values of $n$ for which the determinant has a real value for any choice of $a_{\mu \nu }$ .

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Problem 2617

For what $n$ is the same determinant (see problem 2616) equal to a purely imaginary number?

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Problem 2618

Show that for odd $n$ the same determinant (see problem 2616) equals $A(1 \pm i)$ , where $A$ is real.

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Problem 2619

Prove the equality:

\begin{vmatrix} 1+x_{1}y_{1} & 1+x_{1}y_{2} & \ldots & 1+x_{1}y_{n} \\ 1+x_{2}y_{1} & 1+x_{2}y_{2} & \ldots & 1+x_{2}y_{n} \\ \vdots & \vdots & \vdots & \vdots \\ 1+x_{n}y_{1} & 1+x_{n}y_{2} & \ldots & 1+x_{n}y_{n} \end{vmatrix}=0, \quad n > 2 .

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Problem 2620

Find the value of the determinant

\Delta =\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha _{1} & \beta _{1} & \gamma _{1} \\ \alpha _{2} & \beta _{2} & \gamma _{2} \end{vmatrix},

whose entries are the cosines of the angles between the new axes and the old ones; the coordinate systems are assumed to be rectangular.

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Problem 2621

Show that the determinant

\begin{vmatrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{vmatrix},

whose entries are the projections of three vectors onto the axes, does not change under a transformation of rectangular coordinates.

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Problem 2622

Solve the system of equations

\begin{cases} 2x \hphantom {{}+4y} -z=1, \\ 2x+4y-z=1, \\ -x+8y+3z=2 . \end{cases}

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Problem 2623

Solve the system of equations

\begin{cases} x+y+z=a, \\ x+(1+a)y+z=2a, \\ x+y+(1+a)z=0 . \end{cases}

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Problem 2624

Solve the system of equations

\begin{cases} x+2y+z+t=0, \\ 2x+y+z+2t=0, \\ x+2y+2z+t=0, \\ x+y+z+t=0 . \end{cases}

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Problem 2625

Solve the system of equations

\begin{cases} 5x+4z+2t=3, \\ x-y+2z+t=1, \\ 4x+y+2z=1, \\ x+y+z+t=0 . \end{cases}

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Problem 2626

Solve the system of equations

\begin{cases} x-4y+2z=-1, \\ 2x-3y-z-5t=-7, \\ 3x-7y+z-5t=-8, \\ y-z-t=-1 . \end{cases}

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Problem 2627

From the system of equations

\begin{cases} x+y+2z=0, \\ 2x-y+z=0, \\ x+y+z=t, \\ 2x-2y+z=u \end{cases}

find $u$ as a function of $t$ .

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Problem 2628

Solve the system:

\begin{cases} x_{1}+x_{2}+x_{3}+\ldots +x_{n}=n, \\ x_{1}+2x_{2}+3x_{3}+\ldots +nx_{n}=\dfrac {n(n+1)}{2}, \\ x_{1}+3x_{2}+6x_{3}+\ldots +\dfrac {n(n+1)}{2}x_{n}=\dfrac {n(n+1)(n+2)}{6}, \\ \vdots \\ x_{1}+nx_{2}+\dfrac {n(n+1)}{2}x_{3}+\ldots +\dfrac {n(n+1)\ldots (2n-2)}{1 \cdot 2 \cdot 3 \ldots (n-1)}x_{n}=\dfrac {n(n+1)\ldots (2n-1)}{1 \cdot 2 \cdot 3 \ldots n} . \end{cases}

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Problem 2629

Solve the system:

\begin{cases} x_{2}+x_{3}+\ldots +x_{n-1}+x_{n}=1, \\ x_{1}+x_{3}+\ldots +x_{n-1}+x_{n}=2, \\ x_{1}+x_{2}+\ldots +x_{n-1}+x_{n}=3, \\ \vdots \\ x_{1}+x_{2}+x_{3}+\ldots +x_{n-1}=n . \end{cases}

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Problem 2630

Show that when the determinant $\Delta$ , formed from the $n^{2}$ entries $a_{\mu \nu }$ , equals zero, the proportion $\dfrac {A_{1\mu }}{A_{1\nu }}=\dfrac {A_{2\mu }}{A_{2\nu }}=\cdots =\dfrac {A_{n\mu }}{A_{n\nu }}$ holds, where $A_{kl}$ is the cofactor of the entry $a_{kl}$ .

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Problem 2631

Solve the system of equations:

a_{\nu 1}x_{1}+a_{\nu 2}x_{2}+\ldots +a_{\nu n}x_{n}=\lambda _{\nu }, \quad \nu =1,2,\ldots ,n,

assuming that the array formed from the coefficients of these equations is orthogonal. Having found the solution, show that $A_{\nu \mu }=a_{\nu \mu } \cdot \Delta$ , where $A_{\nu \mu }$ is the cofactor of the entry $a_{\nu \mu }$ .

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Problem 2632

Given

\Delta =\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1p} \\ a_{21} & a_{22} & \ldots & a_{2p} \\ \vdots & \vdots & \vdots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{vmatrix} \neq 0,

and all the determinants of order $p+1$ of the matrix

\begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1m} \\ a_{21} & a_{22} & \ldots & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nm} \end{pmatrix}, \quad m > p, \; n > p,

containing $\Delta$ as a minor are equal to zero. Prove that the rank of the matrix equals $p$ .

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Problem 2633

Under what condition do three points $(x_{1},y_{1})$ , $(x_{2},y_{2})$ , $(x_{3},y_{3})$ lie on a single line?

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Problem 2634

Under what condition do three lines $a_{1}x+b_{1}y+c_{1}=0$ , $a_{2}x+b_{2}y+c_{2}=0$ , $a_{3}x+b_{3}y+c_{3}=0$ intersect at a single point?

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Problem 2635

Under what condition do four points $M_{1}(x_{1},y_{1},z_{1})$ , $M_{2}(x_{2},y_{2},z_{2})$ , $M_{3}(x_{3},y_{3},z_{3})$ , $M_{4}(x_{4},y_{4},z_{4})$ lie in a single plane?

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Problem 2636

Under what conditions do $n$ points $M_{\nu }(x_{\nu },y_{\nu },z_{\nu })$ , $\nu =1,2,\ldots ,n$ , lie in a single plane, and under what conditions do they lie on a single line?

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Problem 2637

Under what conditions do $n$ planes $A_{\nu }x+B_{\nu }y+C_{\nu }z+D_{\nu }=0$ , $\nu =1,2,\ldots ,n$ , intersect at a single point, or in a single line?

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