solve() function solves equation a %*% x = b for x, where b is a vector or matrix.

solve(a, b, tol, LINPACK = FALSE, ...)

• a: coefficients of the equation
• b: vector or matrix of the equation right side
• tol: the tolerance for detecting linear dependencies in the columns of a
• LINPACK: logical. Defunct and ignored

5x = 10, what's x?

>solve(5,10)

[1] 2

Let's see two variables examples:
3x + 2y = 8
x + y =2
What's x and y?

In above equations, matrix a is:
  3 2
  1 1
Matrix b is:
  8
  2

> a <- matrix(c(3,1,2,1),nrow=2,ncol=2)
> a

[,1] [,2]
[1,] 3 2
[2,] 1 1

> b <- matrix(c(8,2),nrow=2,ncol=1)
> b

[,1]
[1,] 8
[2,] 2

> solve(a,b)

[,1]
[1,] 4
[2,] -2

So x = 4, y = -2.

If b is absent, the default is a unit matrix.

> x <- stats::rnorm(16)
> dim(x) <- c(4,4)
> x

[,1] [,2] [,3] [,4]
[1,] -0.3017359 -0.4687800 0.66832626 0.003768864
[2,] -0.8327101 0.7754996 -0.04494932 1.900833149
[3,] -0.1948664 -0.9313664 -0.47685005 -0.123290962
[4,] 1.2502012 -1.0014304 1.61952675 1.119330272

> solve(x)

[,1] [,2] [,3] [,4]
[1,] -1.0175034 -0.23116550 -0.09488446 0.38553721
[2,] -0.2013479 0.03601077 -0.78443594 -0.14687844
[3,] 0.8975934 -0.08140970 -0.59455159 0.06973859
[4,] -0.3423730 0.40820022 0.26440712 0.23046715

Get the inverse matrix of matrix x:

> solve(x) %*% x

[,1] [,2] [,3] [,4]
[1,] 1.000000e+00 0.000000e+00 -2.220446e-16 2.775558e-16
[2,] 8.881784e-16 1.000000e+00 -8.881784e-16 2.220446e-16
[3,] -8.881784e-16 0.000000e+00 1.000000e+00 -4.440892e-16
[4,] 0.000000e+00 -2.775558e-17 2.775558e-17 1.000000e+00