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# How to Find x y and z Values when Inverse Matrix is known?

TopMatrix 'P' can be called an inverse matrix when another matrix 'Q' exists, such that PQ = QP = I.
Here 'I' is identity matrix (n, n). P and Q are also matrices with 'n' row and 'n' column that is (n, n). In an identity matrix, all elements except diagonal will be zero. All elements of diagonal in identity matrix will be ‘1’ and will be represented as:

Here multiplication of matrices is normal matrix multiplication. Matrix 'Q' is determined by 'P' and is called as inverse of 'P'. it is denoted by P − 1. It is essential that matrix must be Square matrix. Non square matrix cannot be inversed.
Let’s try to find out how to find x y and z values when inverse matrix is known. Consider following three equations:
p x + q y + r z = a,
I x +j y + k z = b,
l x +m y + n z = c,
Above can be expressed as a product of matrices as:
=

Lets write this matrix in form [A] [X] = [B] where 'X' and 'B' are called column vector. To find the value of x, y and z, inverse matrix of 'A' is multiplied to either side that is:
(A-1 * A) * X = A-1 * B
Since multiplication of A-1 and A gives an identity matrix (I) and product of I and X gives X. therefore value of 'X' can be calculated as:
X = A-1 * B,
Since we are given inverse A-1, and value of B is also known, hence we can calculate the value of 'X' by multiplying the matrices.
Similarly,
Y = A-1 * B and
Z = A-1 * B.