#### 7.9Solving Systems of Equations

 procedure(matrix-solve M B [fail]) → (U F (Matrix Number)) M : (Matrix Number) B : (Matrix Number) fail : (-> F) = (λ () (error ...))
Returns the matrix X for which (matrix* M X) is B. M and B must have the same number of rows.

It is typical for B (and thus X) to be a column matrix, but not required. If B is not a column matrix, matrix-solve solves for all the columns in B simultaneously.

Examples:
> (define M (matrix [[7 5] [3 -2]]))
> (define B0 (col-matrix [3 22]))
> (define B1 (col-matrix [19 4]))
> (matrix-solve M B0)
 - : #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #)

(array #[#[4] #[-5]])

> (matrix* M (col-matrix [4 -5]))
 - : #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Integer)) # # #)

(array #[#[3] #[22]])

> (matrix-solve M B1)
 - : #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #)

(array #[#[2] #[1]])

> (matrix-cols (matrix-solve M (matrix-augment (list B0 B1))))
 - : (Listof #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #))

(list (array #[#[4] #[-5]]) (array #[#[2] #[1]]))

matrix-solve does not solve overconstrained or underconstrained systems, meaning that M must be invertible. If M is not invertible, the result of applying the failure thunk fail is returned.

matrix-solve is implemented using matrix-gauss-elim to preserve exactness in its output, with partial pivoting for greater numerical stability when M is not exact.

See vandermonde-matrix for an example that uses matrix-solve to compute Legendre polynomials.

 procedure(matrix-inverse M [fail]) → (U F (Matrix Number)) M : (Matrix Number) fail : (-> F) = (λ () (error ...))
Wikipedia: Invertible Matrix Returns the inverse of M if it exists; otherwise returns the result of applying the failure thunk fail.

Examples:
> (matrix-inverse (identity-matrix 3))
 - : #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #)

(array #[#[1 0 0] #[0 1 0] #[0 0 1]])

> (matrix-inverse (matrix [[7 5] [3 -2]]))
 - : #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #)

(array #[#[2/29 5/29] #[3/29 -7/29]])

> (matrix-inverse (matrix [[1 2] [10 20]]))

matrix-inverse: contract violation

expected: matrix-invertible?

given: (array #[#[1 2] #[10 20]])

> (matrix-inverse (matrix [[1 2] [10 20]]) (λ () #f))
 - : (U #(struct:Array (Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real)) # # #) False)

#f

 procedure M : (Matrix Number)
Returns #t when M is a square-matrix? and (matrix-determinant M) is nonzero.

 procedure M : (Matrix Number)
Wikipedia: Determinant Returns the determinant of M, which must be a square-matrix?.

Examples:
 > (matrix-determinant (diagonal-matrix '(1 2 3 4))) - : Real 24 > (* 1 2 3 4) - : Integer [more precisely: Positive-Integer] 24 > (matrix-determinant (matrix [[1 2] [10 20]])) - : Real 0 > (matrix-determinant (col-matrix [1 2])) square-matrix-size: contract violation expected: square-matrix? given: (array #[#[1] #[2]])