类12 NCERT解决方案-数学第I部分–第4章行列式–练习4.6 |套装1

在练习1至6中检查方程组的一致性。

问题1. x + 2y = 2

2x + 3y = 3

解决方案：

Matrix form of the given equations is AX = B

where, A =$\begin{bmatrix}1 & 2 \\2 & 3 \\\end{bmatrix}$ , B = $\begin{bmatrix}2 \\3 \\\end{bmatrix}$  and, X = $\begin{bmatrix}x\\y\\\end{bmatrix}$

∴  $\begin{bmatrix}1 & 2 \\2 & 3 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}2 \\3 \\\end{bmatrix}$

Now, |A| = ${\begin{vmatrix}1&2\\2&3\end{vmatrix}} = 3-4 = -1 ≠ 0$

∵ Inverse of matrix exists, unique solution.

∴ System of equation is consistent.

问题2。2x– y = 5

x + y = 4

解决方案：

Matrix form of the given equations is AX = B

where, A =$\begin{bmatrix}2 & -1 \\1 & 1 \\\end{bmatrix}$, B =$\begin{bmatrix}5 \\4 \\\end{bmatrix}$  and, X =$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴$\begin{bmatrix}2 & -1 \\1 & 1 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}5 \\4 \\\end{bmatrix}$

Now, |A| =${\begin{vmatrix}2&-1\\1&1\end{vmatrix}} = 2-(-1) = 3 ≠ 0$

∵ Inverse of matrix exists, unique solution.

∴ System of equation is consistent.

问题3. x + 3y = 5

2x + 6y = 8

解决方案：

Matrix form of the given equations is AX = B

where, A =$\begin{bmatrix}1 &3 \\2 & 6 \\\end{bmatrix}$, B = $\begin{bmatrix}5 \\8 \\\end{bmatrix}$ and, X =$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴ $\begin{bmatrix}1 & 3 \\2 & 6 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}5 \\8 \\\end{bmatrix}$

Now, |A| =${\begin{vmatrix}1&3\\2&6\end{vmatrix}} = 6-6=0$

And, adj. A =$\begin{bmatrix}6& -3 \\-2& 1 \\\end{bmatrix}$

∴ (adj. A) B = $\begin{bmatrix}6 & -3 \\-2 & 1 \\\end{bmatrix}\begin{bmatrix}5\\8\\\end{bmatrix}=\begin{bmatrix}30-24 \\-10+8 \\\end{bmatrix}=\begin{bmatrix}6\\-2\\\end{bmatrix} ≠0$

∵ Have no common solution.

∴ System of equation is inconsistent.

问题4. x + y + z = 1

2x + 3y + 2z = 2

斧头+ ay + 2az = 4

解决方案：

Matrix form of the given equations is AX = B

where, A =$\begin{bmatrix}1 & 1 & 1\\2 & 3 & 2\\a & a & 2a\end{bmatrix}$, B =$\begin{bmatrix}1\\2\\4\end{bmatrix}$and, X =$\begin{bmatrix}x\\y\\z\end{bmatrix}$

∴  $\begin{bmatrix}1 & 1 & 1\\2 & 3 & 2\\a & a & 2a\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\4\end{bmatrix}$

Now, |A| = ${\begin{vmatrix}1&1&1\\2&3&2\\a&a&2a\end{vmatrix}} = 1(6a-2a)-1(4a-2a)+1(2a-3a)=4a-2a-a=a≠0$

∵ Inverse of matrix exists, unique solution.

∴ System of equation is consistent.

问题5。3x – y – 2z = 2

2y – z = -1

3x – 5y = 3

解决方案：

Matrix form of the given equations is AX = B

where, A =${\begin{bmatrix}-5&10&5\\-3&6&3\\-6&12&6\end{bmatrix}}$, B=$\begin{bmatrix}2\\-1\\3\end{bmatrix}$and, X =$\begin{bmatrix}x\\y\\z\end{bmatrix}$

∴${\begin{bmatrix}-5&10&5\\-3&6&3\\-6&12&6\end{bmatrix}}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}2\\-1\\3\end{bmatrix}$

Now, |A| =${\begin{vmatrix}-5&10&5\\-3&6&3\\-6&12&6\end{vmatrix}} = 3(0-5)-(-1)(0+3)+(-2)(0-6)=3(-5)+3+12=-15+15=0$

And, adj. A =$\begin{bmatrix}-5 & 10 & 5\\-3 & 6 & 3\\-6 & 12 & 6\end{bmatrix}$

∴ (adj. A) B =$\begin{bmatrix}-5 & 10 & 5\\-3 & 6 & 3\\-6 & 12 & 6\end{bmatrix}\begin{bmatrix}2\\-1\\3\end{bmatrix}=\begin{bmatrix}-10-10-15\\-6-6+9 \\-12-12+18\end{bmatrix}=\begin{bmatrix}-5\\-3\\-6\end{bmatrix} ≠0$

∴ System of equation is inconsistent.

问题6. 5x – y + 4z = 5

2x + 3y + 5z = 2

5x – 2y + 6z = –1

解决方案：

Matrix form of the given equations is AX = B

where, A =$\begin{bmatrix}5 & -1 & 4\\2 & 3 & 5\\5 & -2 & 6\end{bmatrix}$, B = $\begin{bmatrix}5\\2\\-1\end{bmatrix}$and, X=$\begin{bmatrix}x\\y\\z\end{bmatrix}$

∴$\begin{bmatrix}5 & -1 & 4\\2 & 3 & 5\\5 & -2 & 6\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}5\\2\\-1\end{bmatrix}$

Now, |A| =$\begin{vmatrix}5 & -1 & 4\\2 & 3 & 5\\5 & -2 & 6\end{vmatrix}=5(8+10)-(-1)(12-25)+4(-4-15)=140-13-76=140-89=51≠0$

∵ Inverse of matrix exists, unique solution.

∴ System of equation is consistent.

在练习7至14中，使用矩阵方法求解线性方程组。

问题7. 5x + 2y = 4

7x + 3y = 5

解决方案：

Matrix form of the given equations is AX = B

where, A=$\begin{bmatrix}5 &2 \\7 & 3 \\\end{bmatrix}$, B=$\begin{bmatrix}4 \\5 \\\end{bmatrix}$, X=$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴$\begin{bmatrix}5 &2 \\7 & 3 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}4 \\5 \\\end{bmatrix}$

Now, |A|=$\begin{vmatrix}5 &2 \\7 & 3 \\\end{vmatrix}=15-14=1 ≠0$

∴Unique solution

Now, X = A^-1B =$\frac{1}{|A|}$(adj.A)B

$\begin{bmatrix}x\\y\\\end{bmatrix}=\frac{1}{1}\begin{bmatrix}3&-2\\-7&5\\\end{bmatrix}\begin{bmatrix}4\\5\\\end{bmatrix}=\begin{bmatrix}12-10 \\-28+25 \\\end{bmatrix}=\begin{bmatrix}2 \\- 3 \\\end{bmatrix}$

Therefore, x=2 and y=-3

问题8。2x – y = -2

3x + 4y = 3

解决方案：

Matrix form of the given equations is AX = B

where, A=$\begin{bmatrix}2 &-1 \\3 & 4 \\\end{bmatrix}$, B=$\begin{bmatrix}-2 \\3 \\\end{bmatrix}$, X=$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴$\begin{bmatrix}2 &-1 \\3 & 4 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}-2 \\3 \\\end{bmatrix}$

Now, |A|=$\begin{vmatrix}2 &-1 \\3 & 4 \\\end{vmatrix}=8-(-3)=8+3=11≠0$

∴Unique solution

Now, X = A^-1B $\frac{1}{|A|}$(adj.A)B

$\begin{bmatrix}x\\y\\\end{bmatrix}=\frac{1}{11}\begin{bmatrix}4&1\\-3&2\\\end{bmatrix}\begin{bmatrix}-2\\3\\\end{bmatrix}=\frac{1}{11}\begin{bmatrix}-8+3 \\6+6 \\\end{bmatrix}=\begin{bmatrix}-5/11 \\12/11 \\\end{bmatrix}$

Therefore, x=-5/11 and y=12/11

问题9. 4x – 3y = 3

3x – 5y = 7

解决方案：

Matrix form of the given equations is AX = B

where, A=$\begin{bmatrix}4 &-3 \\3 & -5 \\\end{bmatrix}$, B=$\begin{bmatrix}3 \\7 \\\end{bmatrix}$, X=$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴$\begin{bmatrix}4 &-3 \\3 & -5 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}3 \\7 \\\end{bmatrix}$

Now, |A|=$\begin{vmatrix}4 &-3 \\3 & -5 \\\end{vmatrix}=-20-(-9)=-20+9=-11≠0$

∴Unique solutio$\begin{vmatrix}4 &-3 \\3 & -5 \\\end{vmatrix}=-20-(-9)=-20+9=-11≠0$n

Now, X =A^-1B $\frac{1}{|A|}$A(adj.A)B

$\begin{bmatrix}x\\y\\\end{bmatrix}=\frac{1}{-11}\begin{bmatrix}-5&3\\-3&4\\\end{bmatrix}\begin{bmatrix}3\\7\\\end{bmatrix}=\frac{1}{-11}\begin{bmatrix}-15+21 \\-9+28 \\\end{bmatrix}=\begin{bmatrix}-6/11 \\-19/11 \\\end{bmatrix}$

Therefore, x= -6/11 and y= -19/11

问题10.5x + 2y = 3

3x + 2y = 5

解决方案：

Matrix form of the given equations is AX = B

where, A=$\begin{bmatrix}5 &2 \\3 & 2 \\\end{bmatrix}$, B=$\begin{bmatrix}3 \\5 \\\end{bmatrix}$, X=$\begin{bmatrix}x\\y\\\end{bmatrix}$

∴$\begin{bmatrix}5 &2 \\3 & 2 \\\end{bmatrix}\begin{bmatrix}x\\y\\\end{bmatrix}=\begin{bmatrix}3 \\5 \\\end{bmatrix}$

Now, |A|=$\begin{vmatrix}5 &2 \\3 & 2 \\\end{vmatrix}=10-6=4≠0$

∴Unique solution

Now, X = A^-1B$\frac{1}{|A|}$A(adj.A)B

$\begin{bmatrix}x\\y\\\end{bmatrix}=\frac{1}{4}\begin{bmatrix}2&-2\\-3&5\\\end{bmatrix}\begin{bmatrix}3\\5\\\end{bmatrix}=\frac{1}{4}\begin{bmatrix}6-10 \\-9+25 \\\end{bmatrix}=\begin{bmatrix}-1 \\4 \\\end{bmatrix}$

Therefore, x= -1 and y= 4

第4章行列式–练习4.6 |套装2