diff --git a/ray-tracer/src/types/matrix.rs b/ray-tracer/src/types/matrix.rs
index 822480e..40257cd 100644
--- a/ray-tracer/src/types/matrix.rs
+++ b/ray-tracer/src/types/matrix.rs
@@ -2,32 +2,25 @@ use crate::types::{eq_f64, Tuple};
 
 #[derive(Clone, Debug)]
 pub struct Matrix {
-    width: usize,
-    height: usize,
+    size: usize,
     values: Vec<f64>,
 }
 
 impl Matrix {
-    pub fn new(width: usize, height: usize) -> Self {
+    pub fn new(size: usize) -> Self {
         Self {
-            width,
-            height,
-            values: vec![0.; width * height],
+            size,
+            values: vec![0.; size * size],
         }
     }
 
     pub fn identity() -> Self {
-        let mut values = vec![0.; 16];
-        values[0] = 1.;
-        values[5] = 1.;
-        values[10] = 1.;
-        values[15] = 1.;
-
-        Self {
-            width: 4,
-            height: 4,
-            values,
-        }
+        Self::from([
+            [1., 0., 0., 0.],
+            [0., 1., 0., 0.],
+            [0., 0., 1., 0.],
+            [0., 0., 0., 1.],
+        ])
     }
 
     pub fn cell(&self, row: usize, column: usize) -> f64 {
@@ -39,11 +32,11 @@ impl Matrix {
         &mut self.values[addr]
     }
 
-    pub fn transpose(self) -> Self {
-        let mut m = Self::new(self.width, self.height);
+    pub fn transpose(&self) -> Self {
+        let mut m = Self::new(self.size);
 
-        for row in 0..self.height {
-            for column in 0..self.width {
+        for row in 0..self.size {
+            for column in 0..self.size {
                 *m.cell_mut(row, column) = self.cell(column, row);
             }
         }
@@ -51,17 +44,97 @@ impl Matrix {
         m
     }
 
+    pub fn invert(&self) -> Self {
+        let mut m = Matrix::new(self.size);
+        let determinant = self.determinant();
+        for row in 0..self.size {
+            for column in 0..self.size {
+                *m.cell_mut(row, column) = self.cofactor(row, column);
+            }
+        }
+
+        let mut m = m.transpose();
+        for row in 0..self.size {
+            for column in 0..self.size {
+                let value = m.cell(row, column);
+                *m.cell_mut(row, column) = value / determinant;
+            }
+        }
+
+        m
+    }
+
+    fn determinant(&self) -> f64 {
+        // TODO: optimization may not be necessary, but this can be optimized by memoizing
+        // submatrices and cofactors.
+        if self.size == 2 {
+            self.cell(0, 0) * self.cell(1, 1) - self.cell(0, 1) * self.cell(1, 0)
+        } else if self.size == 3 {
+            self.cell(0, 0) * self.cofactor(0, 0)
+                + self.cell(0, 1) * self.cofactor(0, 1)
+                + self.cell(0, 2) * self.cofactor(0, 2)
+        } else if self.size == 4 {
+            self.cell(0, 0) * self.cofactor(0, 0)
+                + self.cell(0, 1) * self.cofactor(0, 1)
+                + self.cell(0, 2) * self.cofactor(0, 2)
+                + self.cell(0, 3) * self.cofactor(0, 3)
+        } else {
+            0.
+        }
+    }
+
+    fn invertible(&self) -> bool {
+        !eq_f64(self.determinant(), 0.)
+    }
+
+    fn submatrix(&self, row: usize, column: usize) -> Self {
+        let mut m = Self::new(self.size - 1);
+
+        for r in 0..self.size {
+            for c in 0..self.size {
+                let dest_r = if r < row {
+                    r
+                } else if r > row {
+                    r - 1
+                } else {
+                    continue;
+                };
+                let dest_c = if c < column {
+                    c
+                } else if c > column {
+                    c - 1
+                } else {
+                    continue;
+                };
+                *m.cell_mut(dest_r, dest_c) = self.cell(r, c);
+            }
+        }
+
+        m
+    }
+
+    fn minor(&self, row: usize, column: usize) -> f64 {
+        self.submatrix(row, column).determinant()
+    }
+
+    fn cofactor(&self, row: usize, column: usize) -> f64 {
+        if (row + column) % 2 == 0 {
+            self.minor(row, column)
+        } else {
+            -self.minor(row, column)
+        }
+    }
+
     #[inline]
     fn addr(&self, row: usize, column: usize) -> usize {
-        row * self.width + column
+        row * self.size + column
     }
 }
 
 impl From<[[f64; 2]; 2]> for Matrix {
     fn from(s: [[f64; 2]; 2]) -> Self {
         Self {
-            width: 2,
-            height: 2,
+            size: 2,
             values: s.concat(),
         }
     }
@@ -70,8 +143,7 @@ impl From<[[f64; 2]; 2]> for Matrix {
 impl From<[[f64; 3]; 3]> for Matrix {
     fn from(s: [[f64; 3]; 3]) -> Self {
         Self {
-            width: 3,
-            height: 3,
+            size: 3,
             values: s.concat(),
         }
     }
@@ -80,8 +152,7 @@ impl From<[[f64; 3]; 3]> for Matrix {
 impl From<[[f64; 4]; 4]> for Matrix {
     fn from(s: [[f64; 4]; 4]) -> Self {
         Self {
-            width: 4,
-            height: 4,
+            size: 4,
             values: s.concat(),
         }
     }
@@ -89,7 +160,7 @@ impl From<[[f64; 4]; 4]> for Matrix {
 
 impl PartialEq for Matrix {
     fn eq(&self, rside: &Matrix) -> bool {
-        if self.width != rside.width || self.height != rside.height {
+        if self.size != rside.size {
             return false;
         };
 
@@ -103,12 +174,10 @@ impl PartialEq for Matrix {
 impl std::ops::Mul for Matrix {
     type Output = Matrix;
     fn mul(self, rside: Matrix) -> Matrix {
-        assert_eq!(self.width, 4);
-        assert_eq!(self.height, 4);
-        assert_eq!(rside.width, 4);
-        assert_eq!(rside.height, 4);
+        assert_eq!(self.size, 4);
+        assert_eq!(rside.size, 4);
 
-        let mut m = Matrix::new(self.width, self.height);
+        let mut m = Matrix::new(self.size);
         for row in 0..4 {
             for column in 0..4 {
                 *m.cell_mut(row, column) = self.cell(row, 0) * rside.cell(0, column)
@@ -125,8 +194,7 @@ impl std::ops::Mul for Matrix {
 impl std::ops::Mul<Tuple> for Matrix {
     type Output = Tuple;
     fn mul(self, rside: Tuple) -> Tuple {
-        assert_eq!(self.width, 4);
-        assert_eq!(self.height, 4);
+        assert_eq!(self.size, 4);
 
         let mut t = [0.; 4];
 
@@ -266,4 +334,103 @@ mod tests {
 
         assert_eq!(a.transpose(), expected);
     }
+
+    #[test]
+    fn calculates_2x2_determinant() {
+        let m = Matrix::from([[1., 5.], [-3., 2.]]);
+        assert_eq!(m.determinant(), 17.);
+    }
+
+    #[test]
+    fn calculates_3x3_determinant() {
+        let m = Matrix::from([[1., 2., 6.], [-5., 8., -4.], [2., 6., 4.]]);
+        assert_eq!(m.cofactor(0, 0), 56.);
+        assert_eq!(m.cofactor(0, 1), 12.);
+        assert_eq!(m.cofactor(0, 2), -46.);
+        assert_eq!(m.determinant(), -196.);
+    }
+
+    #[test]
+    fn calculates_4x4_determinant() {
+        let m = Matrix::from([
+            [-2., -8., 3., 5.],
+            [-3., 1., 7., 3.],
+            [1., 2., -9., 6.],
+            [-6., 7., 7., -9.],
+        ]);
+        assert_eq!(m.cofactor(0, 0), 690.);
+        assert_eq!(m.cofactor(0, 1), 447.);
+        assert_eq!(m.cofactor(0, 2), 210.);
+        assert_eq!(m.cofactor(0, 3), 51.);
+        assert_eq!(m.determinant(), -4071.);
+    }
+
+    #[test]
+    fn calculates_submatrix() {
+        let m = Matrix::from([[1., 5., 0.], [-3., 2., 7.], [0., 6., -3.]]);
+        let expected = Matrix::from([[-3., 2.], [0., 6.]]);
+        assert_eq!(m.submatrix(0, 2), expected);
+
+        let m = Matrix::from([
+            [-6., 1., 1., 6.],
+            [-8., 5., 8., 6.],
+            [-1., 0., 8., 2.],
+            [-7., 1., -1., 1.],
+        ]);
+        let expected = Matrix::from([[-6., 1., 6.], [-8., 8., 6.], [-7., -1., 1.]]);
+        assert_eq!(m.submatrix(2, 1), expected);
+    }
+
+    #[test]
+    fn calculates_minors_and_cofactors() {
+        let m = Matrix::from([[3., 5., 0.], [2., -1., -7.], [6., -1., 5.]]);
+        assert_eq!(m.submatrix(1, 0).determinant(), 25.);
+        assert_eq!(m.minor(0, 0), -12.);
+        assert_eq!(m.cofactor(0, 0), -12.);
+        assert_eq!(m.minor(1, 0), 25.);
+        assert_eq!(m.cofactor(1, 0), -25.);
+    }
+
+    #[test]
+    fn invert_4x4_matrix() {
+        let m = Matrix::from([
+            [6., 4., 4., 4.],
+            [5., 5., 7., 6.],
+            [4., -9., 3., -7.],
+            [9., 1., 7., -6.],
+        ]);
+
+        assert_eq!(m.determinant(), -2120.);
+        assert!(m.invertible());
+
+        let m = Matrix::from([
+            [-4., 2., -2., -3.],
+            [9., 6., 2., 6.],
+            [0., -5., 1., -5.],
+            [0., 0., 0., 0.],
+        ]);
+        assert_eq!(m.determinant(), 0.);
+        assert!(!m.invertible());
+
+        let m = Matrix::from([
+            [-5., 2., 6., -8.],
+            [1., -5., 1., 8.],
+            [7., 7., -6., -7.],
+            [1., -3., 7., 4.],
+        ]);
+        let expected = Matrix::from([
+            [0.21805, 0.45113, 0.24060, -0.04511],
+            [-0.80827, -1.45677, -0.44361, 0.52068],
+            [-0.07895, -0.22368, -0.05263, 0.19737],
+            [-0.52256, -0.81391, -0.30075, 0.30639],
+        ]);
+
+        assert_eq!(m.determinant(), 532.);
+        assert_eq!(m.cofactor(2, 3), -160.);
+        assert!(eq_f64(expected.cell(3, 2), -160. / 532.));
+        assert_eq!(m.cofactor(3, 2), 105.);
+        assert!(eq_f64(expected.cell(2, 3), 105. / 532.));
+
+        assert_eq!(m.invert(), expected);
+    }
 }