Q. x1+x2+2x3=−1x1−2x2+x3=−53x1+x2+x3=3Find all solutions by nsing the Gaussian elimination \& Gauss-Jordan Reduction
Write Augmented Matrix: Write the augmented matrix for the system of equations.⎣⎡1131−21211∣∣∣−1−53⎦⎤
Leading 1 in First Column: Perform row operations to get a leading 1 in the first row, first column (R1 is already set).No changes needed for R1.
Make Zeros Below Leading 1: Make the elements below the leading 1 in the first column zero, using R2 - R1 → R2 and 3R1 - R3 → R3.⎣⎡1001−3−22−1−5∣∣∣−1−46⎦⎤
Leading 1 in Second Column: Make the leading coefficient in the second row, second column a 1 by multiplying R2 by −1/3.⎣⎡10011−221/3−5∣∣∣−14/36⎦⎤
Eliminate Second Column: Eliminate the second column below and above the leading 1 in R2. Add 2R2 to R3 and subtract R2 from R1.⎣⎡1000105/31/3−13/3∣∣∣−7/34/314/3⎦⎤
Leading 1 in Third Column: Make the leading coefficient in the third row, third column a 1 by multiplying R3 by −3/13.⎣⎡1000105/31/31∣∣∣−7/34/3−14/13⎦⎤
Eliminate Third Column: Eliminate the third column above R3 by subtracting (5/3)R3 from R1 and subtracting (1/3)R3 from R2.⎣⎡100010001∣∣∣12−14/13⎦⎤