Using the inverse matrix method, solve the system of...

Using the inverse matrix method, solve the system of equations for each of the extbf{b} matrices. $x_1 + 5x_2 - 3x_3 = b_1$ $3x_1 + x_2 + 5x_3 = b_2$ $x_1 + 3x_2 - 2x_3 = b_3$ $egin{bmatrix} b_1 \ b_2 \ b_3 end{bmatrix} = egin{bmatrix} 5 \ 1 \ 2 end{bmatrix}$ $(x_1, x_2, x_3) = oxed{} $egin{bmatrix} b_1 \ b_2 \ b_3 end{bmatrix} = egin{bmatrix} -4 \ 16 \ -12 end{bmatrix}$ $(x_1, x_2, x_3) = oxed{} $egin{bmatrix} b_1 \ b_2 \ b_3 end{bmatrix} = egin{bmatrix} -3 \ 19 \ -7 end{bmatrix}$ $(x_1, x_2, x_3) = oxed{}

Transcript

00:02 Hello, in this question we have to use the inverse matrix method to solve the system of equations for each of the b matrices.

00:13 So we are given this equation, this, we are given a system of linear equations which we have written down as a matrix equation.

00:24 This which is equal to this matrix times x1, x2, x3 equal to b1, b2, b3.

00:30 Now we have to solve this equation for three different values of b vector.

00:38 The first b vector is 512, second b vector is minus 416 minus 12 and the third b vector is minus 319 minus 7.

00:47 Okay, now we are given this equation, linear system of linear equations ax which is equivalent to the matrix equation ax equal to b.

00:59 Now if we multiply a inverse to both sides of this matrix equation we get a inverse times a is equal to x.

01:07 A inverse times a times x is equal to a inverse times b.

01:11 But a inverse times a is identity, identity times x is x.

01:16 So we get the equation x equal to a inverse times b.

01:21 Now, so if a is invertible and we can find its a inverse then this says that the solution of this equation must be equal to the inverse of a multiplied to b.

01:36 So to, and this is the matrix, inverse matrix method of solving a system of linear equations and to solve this first we have to find a inverse.

01:48 We are going to use gauss jordan elimination to find a inverse.

01:51 So we start with the given matrix a and we construct an augmented matrix ai and then we are going to apply row operations to reduce this matrix to identity.

02:05 At the same time we will also apply the same operations to the matrix that we have on the right and finally when this matrix, the left hand side matrix will be reduced to identity, the right hand side matrix will become the a inverse.

02:21 Okay, so we start this process with a and identity...

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