How do you find the inverse of A in system of linear equation and matrices?

Answer 1

First, we need to recall that a linear equation does not involve any products or roots of variables. All variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions. For example, x + √y = 4, y = sin x, and 2x + y - z + yz = 5 are not linear.

To solve a system of equations such as

3x + y = 5

2x - y = 3

all information required for the solution is emboded in the augmented matrix (imagine that I put those information into a rectangular arrays)

3 1 5

2 -1 3

and that the solution can be obtained by performing appropriate operations on this matrix.

The matrix of this system linear equations is a square matrix A such as

3 1

2 -1

Think this matrix as

a b

c d

To find an inverse of this square matrix A (2 x 2), we need to find a matrix B of the same size such that AB = I and BA = I, then A is said to be invertible and B is called the inverse of A. If no such a matrix can be found, then A is said to be singular.

An invertible matrix has exactly one inverse.

A square matrix A is invertible if ad - bc ≠ 0 (where ad - bc is the determinant)

The formula of finding the inverse of a square matrix A is

A-1 = [1/(ad - bc)][d -b the second row -c a](I'm sorry, I can't draw the arrays)

So let's find the inverse of our example.

A-1 = [1/(-3 -2)][-1 -1 the second row -2 3] = [-1/-5 -1/-5 the sec. row -2/-5 3/-5] =

1/5 1/5

2/5 -3/5

A n x m matrix cannot have an inverse. A n x n matrix may or may not have an inverse.

To find the inverse of a n x n matrix we should to adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A | I]. Then we should apply row opperations to this matrix until the left side is reduced to I. This opperations will convert the right side to A-1, so the final matrix will have the form [I |A-1 ].

(There are many other methods how to find the inverse of a n x n matrix, but I can't show them by examples. I am so sorry that I can't be so much useful to you).