Using Elementary Row Operations, Find the Inverse of the Matrix a = ⎛ ⎜ ⎝ 3 − 3 4 2 − 3 4 0 − 1 1 ⎞ ⎟ ⎠ and Hence Solve the Following System of Equations - Mathematics

Sum

Using elementary row operations, find the inverse of the matrix A = ((3, 3,4),(2,-3,4),(0,-1,1)) and hence solve the following system of equations :  3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.

Solution

Here A = ((3,-3,4),(2,-3,4),(0,-1,1))

Using elementary row operations, "A" = "I.A"

⇒ ((3,-3,4),(2,-3,4),(0,-1,1)) = ((1,0,0),(0,1,0),(0,-1,1))"A"           ..."By"   "R"_1->"R"_1 -"R"_2

⇒ ((1,0,0),(2,-3,4),(0,-1,1)) = ((1,-1,0),(0,1,0),(0,0,1))"A"           ..."By"   "R"_2->"R"_2 -2"R"_1

⇒ ((1,0,0),(0,-3,4),(0,-1,1)) = ((1,-1,0),(-2,3,0),(0,0,1))"A"           ..."By"   "R"_2->"R"_2 -4"R"_3

⇒ ((1,0,0),(0,1,0),(0,-1,1)) = ((1,-1,0),(-2,3,-4),(0,0,1))"A"           ..."By"   "R"_3->"R"_3 +"R"_2

⇒ ((1,0,0),(0,1,0),(0,0,1)) = ((1,-1,0),(-2,3,-4),(-2,3,-3))"A"

Since we know that "I" = "A"^-1 "A" "so", "A"^-1 = ((1,-1,0),(-2,3,-4),(-2,3,-3)).

Now consider the equations 3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.

The matrix form of these equations is, ((3,-3,4),(2,-3,4),(0,-1,1)) (("x"),("y"),("z")) = ((21),(20),(5))

Where A = ((3,-3,4),(2,-3,4),(0,-1,1)), "X" = (("x"),("y"),("z")) and "B" = ((21),(20),(5))

So, "AX" = "B"

⇒ "X" = "A"^-1 "B"

⇒ "X" = ((1,-1,0),(-2,3,-4),(-2,3,-3))((21),(20),(5))

⇒ (("x"),("y"),("z"))  = ((1),(-2),(3))

Therefore, x = 1, y = -2, Z = 3.

Concept: Elementary Operation (Transformation) of a Matrix
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