Answer to Question #188651 in Algebra for Nima wangdi

Question #188651

If a= [4,3]

[2,5] 2×2 matrix.

Find x and y such that A²-xA + y I = 0


1
Expert's answer
2021-05-07T11:32:54-0400

"Given \\begin{bmatrix}\n 4 & 3 \\\\\n 2 & 5\n\\end{bmatrix}_{2\\times 2}"


"A^2-xA+yI=0.....................................................(i)"


"A^2=A.A=\\begin{bmatrix}\n 4 & 3\\\\\n 2 & 5\n\\end{bmatrix}\\begin{bmatrix}\n 4 & 3 \\\\\n 2 & 5\n\\end{bmatrix}=\\begin{bmatrix}\n 16+6 & 12+15 \\\\\n 8+10 & 6+25\n\\end{bmatrix}"


"A^2=\\begin{bmatrix}\n 22 & 27 \\\\\n 18 & 31\n\\end{bmatrix}"

Value of A2 and A put in equation (i). I is identity matrix.


"\\begin{bmatrix}\n 22 & 27 \\\\\n 18 & 31\n\\end{bmatrix}-x\\begin{bmatrix}\n 4 & 3 \\\\\n 2 & 5\n\\end{bmatrix}+y\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{bmatrix}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 & 0\n\\end{bmatrix}"


"\\begin{bmatrix}\n 22 & 27 \\\\\n 18 & 31\n\\end{bmatrix}-\\begin{bmatrix}\n 4x & 3x \\\\\n 2x & 5x\n\\end{bmatrix}+\\begin{bmatrix}\n y & 0 \\\\\n 0 & y\n\\end{bmatrix}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 & 0\n\\end{bmatrix}"


"\\begin{bmatrix}\n 22-4x+y & 27-3x \\\\\n 18-2x & 31-5x+y\n\\end{bmatrix}=\\begin{bmatrix}\n 0 & 0 \\\\\n 0 & 0\n\\end{bmatrix}"


"22-4xy+y=0.......................................................(ii)"


"27-3x=0"

"3x=27"

"x=9"



"18-2x=0"

"2x=18"

"x=9"


"31-5x+y=0" putting the value of x in this equation,

"31-(5\\times 9)+y=0"

"y=14"



"x=9," "y=14"

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