> a <- matrix(1:12,2,6)
> x <- matrix(1:12,6,2)
> a[,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    3    5    7    9   11
[2,]    2    4    6    8   10   12
> x[,1] [,2]
[1,]    1    7
[2,]    2    8
[3,]    3    9
[4,]    4   10
[5,]    5   11
[6,]    6   12
> b <- a %*% x
> b[,1] [,2]
[1,]  161  377
[2,]  182  434
> solve(a,b)
Error in solve.default(a, b) : 'a' (2 x 6) must be square

   F77_CALL(dgesv)(&n, &p, avals, &n, ipiv, REAL(B), &n, &info);if (info < 0)error(_("argument %d of Lapack routine %s had invalid value"),-info, "dgesv");if (info > 0)error(_("Lapack routine %s: system is exactly singular: U[%d,%d] = 0"),"dgesv", info, info);

错误举例 :

> a <- matrix(1:16,4,4)
> solve(a, a %*% a)
Error in solve.default(a, a %*% a) : Lapack routine dgesv: system is exactly singular: U[3,3] = 0

> a <- matrix(1:4,2,2)
> a[,1] [,2]
[1,]    1    3
[2,]    2    4
> solve(a)   # solve(a) , 不给b的话, 其实b 默认就是和a维度一致并且对角线为1的矩阵.[,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5
> solve(a, diag(1,2,2))   # 可以看到[,1] [,2]
[1,]   -2  1.5
[2,]    1 -0.5
> diag(1,2,2)[,1] [,2]
[1,]    1    0
[2,]    0    1
> a %*% solve(a)   #  因为solve(a,b) = x, a %*% x = b, 所以 a %*% solve(a) = b = diag(1,2,2)[,1] [,2]
[1,]    1    0
[2,]    0    1

b还可以是向量, 向量会自动转成矩阵, 例如

> solve(a, c(2,3))
[1] 0.5 0.5
> a %*% solve(a, c(2,3))[,1]
[1,]    2
[2,]    3

[参考]

solve                   package:base                   R DocumentationSolve a System of EquationsDescription:This generic function solves the equation ‘a %*% x = b’ for ‘x’,where ‘b’ can be either a vector or a matrix.Usage:solve(a, b, ...)## Default S3 method:solve(a, b, tol, LINPACK = FALSE, ...)Arguments:a: a square numeric or complex matrix containing thecoefficients of the linear system.  Logical matrices arecoerced to numeric.b: a numeric or complex vector or matrix giving the right-handside(s) of the linear system.  If missing, ‘b’ is taken to bean identity matrix and ‘solve’ will return the inverse of‘a’.tol: the tolerance for detecting linear dependencies in thecolumns of ‘a’.  The default is ‘.Machine$double.eps’. Notcurrently used with complex matrices ‘a’.LINPACK: logical.  Defunct and ignored....: further arguments passed to or from other methods

/* Real case of solve.default */
static SEXP La_solve(SEXP A, SEXP Bin, SEXP tolin)
{int n, p;double *avals, anorm, rcond, tol = asReal(tolin), *work;SEXP B, Adn, Bdn;if (!(isMatrix(A) && (TYPEOF(A) == REALSXP || TYPEOF(A) == INTSXP || TYPEOF(A) == LGLSXP)))error(_("'a' must be a numeric matrix"));int *Adims = INTEGER(coerceVector(getAttrib(A, R_DimSymbol), INTSXP));n = Adims[0];if(n == 0) error(_("'a' is 0-diml"));int n2 = Adims[1];if(n2 != n) error(_("'a' (%d x %d) must be square"), n, n2);Adn = getAttrib(A, R_DimNamesSymbol);if (isMatrix(Bin)) {int *Bdims = INTEGER(coerceVector(getAttrib(Bin, R_DimSymbol), INTSXP));p = Bdims[1];if(p == 0) error(_("no right-hand side in 'b'"));int p2 = Bdims[0];if(p2 != n)error(_("'b' (%d x %d) must be compatible with 'a' (%d x %d)"),p2, p, n, n);PROTECT(B = allocMatrix(REALSXP, n, p));SEXP Bindn =  getAttrib(Bin, R_DimNamesSymbol);// This is somewhat odd, but Matrix relies on dropping NULL dimnamesif (!isNull(Adn) || !isNull(Bindn)) {// rownames(ans) = colnames(A), colnames(ans) = colnames(Bin)Bdn = allocVector(VECSXP, 2);if (!isNull(Adn)) SET_VECTOR_ELT(Bdn, 0, VECTOR_ELT(Adn, 1));if (!isNull(Bindn)) SET_VECTOR_ELT(Bdn, 1, VECTOR_ELT(Bindn, 1));if (!isNull(VECTOR_ELT(Bdn, 0)) || !isNull(VECTOR_ELT(Bdn, 1)))setAttrib(B, R_DimNamesSymbol, Bdn);}} else {p = 1;if(length(Bin) != n)error(_("'b' (%d x %d) must be compatible with 'a' (%d x %d)"),length(Bin), p, n, n);        PROTECT(B = allocVector(REALSXP, n));if (!isNull(Adn)) setAttrib(B, R_NamesSymbol, VECTOR_ELT(Adn, 1));}PROTECT(Bin = coerceVector(Bin, REALSXP));Memcpy(REAL(B), REAL(Bin), (size_t)n * p);int *ipiv = (int *) R_alloc(n, sizeof(int));/* work on a copy of A */if (!isReal(A)) {A = coerceVector(A, REALSXP);avals = REAL(A);} else {avals = (double *) R_alloc((size_t)n * n, sizeof(double));Memcpy(avals, REAL(A), (size_t)n * n);}PROTECT(A);int info;F77_CALL(dgesv)(&n, &p, avals, &n, ipiv, REAL(B), &n, &info);if (info < 0)error(_("argument %d of Lapack routine %s had invalid value"),-info, "dgesv");if (info > 0)error(_("Lapack routine %s: system is exactly singular: U[%d,%d] = 0"),"dgesv", info, info);if(tol > 0) {char one[2] = "1";anorm = F77_CALL(dlange)(one, &n, &n, REAL(A), &n, (double*) NULL);work = (double *) R_alloc(4*(size_t)n, sizeof(double));F77_CALL(dgecon)(one, &n, avals, &n, &anorm, &rcond, work, ipiv, &info);if (rcond < tol)error(_("system is computationally singular: reciprocal condition number = %g"),rcond);}UNPROTECT(3); /* B, Bin, A */return B;
}