Answer to Question #345264 in Algebra for Geoffroy yawogan

Question #345264

1. Solve the matrix equation 4π’³βˆ’β„¬=𝒳ℬ+2π’œ if π’œ=(βˆ’2141), ℬ=(71βˆ’72).

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2. Calculate the inverse matrix to the matrix π’œ=(001011111). Check whether the obtained inverse matrix is correct.

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3. Solve the system of linear equations: π‘₯βˆ’2𝑦+𝑧=5,βˆ’2π‘₯+3π‘¦βˆ’π‘§=βˆ’8,βˆ’π‘₯βˆ’π‘¦+2𝑧=2.

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4. Calculate the area of the flat shape bounded by the curves: 𝑦=√π‘₯βˆ’1,𝑦=3βˆ’π‘₯,𝑦=0.

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5. Find all the extrema of the function 𝑓(π‘₯)=√16βˆ’π‘₯2.

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6. Find the maximal intervals of convexity (concavity) of the function 𝑓(π‘₯)=2π‘₯+arctg(3π‘₯). Find the respective inflection points.

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1
Expert's answer
2022-05-27T13:59:48-0400

1.


"4X-B=XB+2A""A=\\begin{pmatrix}\n -2 & 1 \\\\\n 4 & 1\n\\end{pmatrix}, B=\\begin{pmatrix}\n 7 & 1 \\\\\n -7 & 2\n\\end{pmatrix}"

Let



"X=\\begin{pmatrix}\n x_{11}& x_{12} \\\\\n x_{21} & x_{22}\n\\end{pmatrix}"

Then



"4X=\\begin{pmatrix}\n 4x_{11}& 4x_{12} \\\\\n 4x_{21} & 4x_{22}\n\\end{pmatrix}""XB=\\begin{pmatrix}\n x_{11}& x_{12} \\\\\n x_{21} & x_{22}\n\\end{pmatrix}\\begin{pmatrix}\n 7 & 1 \\\\\n -7 & 2\n\\end{pmatrix}""=\\begin{pmatrix}\n 7x_{11}-7x_{12}& x_{11}+2x_{12} \\\\\n 7x_{21}-7x_{22} & x_{21}+2x_{22}\n\\end{pmatrix}""4X-BX=\\begin{pmatrix}\n -3x_{11}+7x_{12}& -x_{11}+2x_{12} \\\\\n -3x_{21}+7x_{22} & -x_{21}+2x_{22}\n\\end{pmatrix}""B+2A=\\begin{pmatrix}\n 3 & 3 \\\\\n 1 & 4\n\\end{pmatrix}""\\begin{pmatrix}\n -3x_{11}+7x_{12}& -x_{11}+2x_{12} \\\\\n -3x_{21}+7x_{22} & -x_{21}+2x_{22}\n\\end{pmatrix}=\\begin{pmatrix}\n 3 & 3 \\\\\n 1 & 4\n\\end{pmatrix}"




"-3x_{11}+7x_{12}=3""-x_{11}+2x_{12}=3"




"-3x_{21}+7x_{22}=1""-x_{21}+2x_{22}=4"




"x_{12}=-6""-x_{11}+2x_{12}=3"




"x_{22}=-11""-x_{21}+2x_{22}=4"




"x_{11}=-21, x_{12}=-6, x_{22}=-11, x_{21}=-26"




"X=\\begin{pmatrix}\n -21& -6 \\\\\n -26 & -11\n\\end{pmatrix}"


2.


"A=\\begin{pmatrix}\n 0 & 0 & 1\\\\\n 0 & 1 & 1\\\\\n 1 & 1 & 1\\\\\n\\end{pmatrix}"

"\\begin{pmatrix}\n 0 & 0 & 1 & | & 1 & 0 & 0\\\\\n 0 & 1 & 1 & | & 0 & 1 & 0\\\\\n 1 & 1 & 1 & | & 0 & 0 & 1\\\\\n\\end{pmatrix}"


Swap the rowsΒ 1andΒ 3


"\\begin{pmatrix}\n 1 & 1 & 1 & | & 0 & 0 & 1\\\\\n 0 & 1 & 1 & | & 0 & 1 & 0\\\\\n 0 & 0 & 1 & | & 1 & 0 & 0\\\\\n\\end{pmatrix}"


"R_1=R_1-R_2"


"\\begin{pmatrix}\n 1 & 0 & 0 & | & 0 & -1 & 1\\\\\n 0 & 1 & 1 & | & 0 & 1 & 0\\\\\n 0 & 0 & 1 & | & 1 & 0 & 0\\\\\n\\end{pmatrix}"

"R_2=R_2-R_3"


"\\begin{pmatrix}\n 1 & 0 & 0 & | & 0 & -1 & 1\\\\\n 0 & 1 & 0 & | & -1 & 1 & 0\\\\\n 0 & 0 & 1 & | & 1 & 0 & 0\\\\\n\\end{pmatrix}"

On the left is the identity matrix. On the right is the inverse matrix.


"A^{-1}=\\begin{pmatrix}\n 0 & -1 & 1\\\\\n -1 & 1 & 0\\\\\n 1 & 0 & 0\\\\\n\\end{pmatrix}"

Check


"AA^{-1}=\\begin{pmatrix}\n 0 & 0 & 1\\\\\n 0 & 1 & 1\\\\\n 1 & 1 & 1\\\\\n\\end{pmatrix}\\begin{pmatrix}\n 0 & -1 & 1\\\\\n -1 & 1 & 0\\\\\n 1 & 0 & 0\\\\\n\\end{pmatrix}"

"=\\begin{pmatrix}\n 0+0+1 & 0+0+0 & 0+0+0\\\\\n 0-1+1 & 0+1+0 & 0+0+0\\\\\n 0-1+1 & -1+1+0 & 1+0+0\\\\\n\\end{pmatrix}"

"=\\begin{pmatrix}\n 1 & 0 & 0\\\\\n 0 & 1 & 0\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}=I_3"

"A^{-1}=\\begin{pmatrix}\n 0 & -1 & 1\\\\\n -1 & 1 & 0\\\\\n 1 & 0 & 0\\\\\n\\end{pmatrix}"

3.


"x-2y+z=5\\\\\n-2x+3y-z=-8\\\\\n-x-y+2z=2"


"-x+y=-3\\\\\n-3y+3z=7\\\\\n-x-y+2z=2"


"x=y+3\\\\\nz=y+\\dfrac{7}{3}\\\\\n-y-3-y+2y+\\dfrac{14}{3}=2"



"x=y+13\\\\\nz=y+1\\\\\n\\dfrac{5}{3}=2"

No solution


4.


"\\sqrt{x-1}=0=>x=1"

"3-x=0=>x=3"

"\\sqrt{x-1}=3-x=>x-1=9-6x+x^2, 1\\le x\\le3"

"x^2-7x+10=0"

"(x-2)(x-5)=0"

Since "1\\le x\\le3," we take "x=2."


"A=\\displaystyle\\int_{1}^{2}\\sqrt{x-1}dx+\\displaystyle\\int_{2}^{3}(3-x)dx"

"=[\\dfrac{2(x-1)^{3\/2}}{3}]\\begin{matrix}\n 2 \\\\\n 1\n\\end{matrix}+[3x-\\dfrac{x^2}{2}]\\begin{matrix}\n 3 \\\\\n2\n\\end{matrix}"

"=\\dfrac{2}{3}-0+9-\\dfrac{9}{2}-6+2=\\dfrac{7}{6}({units}^2)"

5.


"16-x^2\\ge0=>-4\\le x\\le 4"

Domain: "[-4, 4]"


"f'(x)=\\dfrac{-2x}{2\\sqrt{16-x^2}}=-\\dfrac{x}{\\sqrt{16-x^2}}"

"f'(x)=0=>-\\dfrac{x}{\\sqrt{16-x^2}}=0"

"x=0"

Critical numbers:"-4, 0, 4."


"f(-4)=0=f(4)"

"f(0)=4"

The function has a local minimum at "(-4,0)" and at "(4,0)."

The function has a local maximum at "(0,4)."

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6.

Domain: "(-\\infin, \\infin)"


"f'(x)=2+\\dfrac{3}{1+9x^2}"

"f''(x)=-\\dfrac{54x}{(1+9x^2)^2}"

"f''(x)=0=>-\\dfrac{54x}{(1+9x^2)^2}=0"

"x=0"

"f(0)=0"

If "x<0, f''(x)>0, f(x)" is concave upward.

If "x>0, f''(x)<0, f(x)" is concave downward.

The function "f(x)" is concave upward on "(-\\infin, 0)."

The function "f(x)" is concave downward on "(0,\\infin)."

Point "(0,0)" is the inflection point.



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