diff --git a/core/math/projection.cpp b/core/math/projection.cpp
index d0ca7c56849..4a0faef08f1 100644
--- a/core/math/projection.cpp
+++ b/core/math/projection.cpp
@@ -596,101 +596,229 @@ Projection Projection::inverse() const {
 }
 
 void Projection::invert() {
-	int i, j, k;
-	int pvt_i[4], pvt_j[4]; /* Locations of pivot matrix */
-	real_t pvt_val; /* Value of current pivot element */
-	real_t hold; /* Temporary storage */
-	real_t determinant = 1.0f;
-	for (k = 0; k < 4; k++) {
-		/** Locate k'th pivot element **/
-		pvt_val = columns[k][k]; /** Initialize for search **/
-		pvt_i[k] = k;
-		pvt_j[k] = k;
-		for (i = k; i < 4; i++) {
-			for (j = k; j < 4; j++) {
-				if (Math::abs(columns[i][j]) > Math::abs(pvt_val)) {
-					pvt_i[k] = i;
-					pvt_j[k] = j;
-					pvt_val = columns[i][j];
-				}
-			}
-		}
+	// Adapted from Mesa's `src/util/u_math.c` `util_invert_mat4x4`.
+	// MIT licensed. Copyright 2008 VMware, Inc. Authored by Jacques Leroy.
+	Projection temp;
+	real_t *out = (real_t *)temp.columns;
+	real_t *m = (real_t *)columns;
 
-		/** Product of pivots, gives determinant when finished **/
-		determinant *= pvt_val;
-		if (Math::is_zero_approx(determinant)) {
-			return; /** Matrix is singular (zero determinant). **/
-		}
+	real_t wtmp[4][8];
+	real_t m0, m1, m2, m3, s;
+	real_t *r0, *r1, *r2, *r3;
 
-		/** "Interchange" rows (with sign change stuff) **/
-		i = pvt_i[k];
-		if (i != k) { /** If rows are different **/
-			for (j = 0; j < 4; j++) {
-				hold = -columns[k][j];
-				columns[k][j] = columns[i][j];
-				columns[i][j] = hold;
-			}
-		}
+#define MAT(m, r, c) (m)[(c) * 4 + (r)]
 
-		/** "Interchange" columns **/
-		j = pvt_j[k];
-		if (j != k) { /** If columns are different **/
-			for (i = 0; i < 4; i++) {
-				hold = -columns[i][k];
-				columns[i][k] = columns[i][j];
-				columns[i][j] = hold;
-			}
-		}
+	r0 = wtmp[0];
+	r1 = wtmp[1];
+	r2 = wtmp[2];
+	r3 = wtmp[3];
 
-		/** Divide column by minus pivot value **/
-		for (i = 0; i < 4; i++) {
-			if (i != k) {
-				columns[i][k] /= (-pvt_val);
-			}
-		}
+	r0[0] = MAT(m, 0, 0);
+	r0[1] = MAT(m, 0, 1);
+	r0[2] = MAT(m, 0, 2);
+	r0[3] = MAT(m, 0, 3);
+	r0[4] = 1.0;
+	r0[5] = 0.0;
+	r0[6] = 0.0;
+	r0[7] = 0.0;
 
-		/** Reduce the matrix **/
-		for (i = 0; i < 4; i++) {
-			hold = columns[i][k];
-			for (j = 0; j < 4; j++) {
-				if (i != k && j != k) {
-					columns[i][j] += hold * columns[k][j];
-				}
-			}
-		}
+	r1[0] = MAT(m, 1, 0);
+	r1[1] = MAT(m, 1, 1);
+	r1[2] = MAT(m, 1, 2);
+	r1[3] = MAT(m, 1, 3);
+	r1[5] = 1.0;
+	r1[4] = 0.0;
+	r1[6] = 0.0;
+	r1[7] = 0.0;
 
-		/** Divide row by pivot **/
-		for (j = 0; j < 4; j++) {
-			if (j != k) {
-				columns[k][j] /= pvt_val;
-			}
-		}
+	r2[0] = MAT(m, 2, 0);
+	r2[1] = MAT(m, 2, 1);
+	r2[2] = MAT(m, 2, 2);
+	r2[3] = MAT(m, 2, 3);
+	r2[6] = 1.0;
+	r2[4] = 0.0;
+	r2[5] = 0.0;
+	r2[7] = 0.0;
 
-		/** Replace pivot by reciprocal (at last we can touch it). **/
-		columns[k][k] = 1.0 / pvt_val;
+	r3[0] = MAT(m, 3, 0);
+	r3[1] = MAT(m, 3, 1);
+	r3[2] = MAT(m, 3, 2);
+	r3[3] = MAT(m, 3, 3);
+
+	r3[7] = 1.0;
+	r3[4] = 0.0;
+	r3[5] = 0.0;
+	r3[6] = 0.0;
+
+	/* choose pivot - or die */
+	if (Math::abs(r3[0]) > Math::abs(r2[0])) {
+		SWAP(r3, r2);
+	}
+	if (Math::abs(r2[0]) > Math::abs(r1[0])) {
+		SWAP(r2, r1);
+	}
+	if (Math::abs(r1[0]) > Math::abs(r0[0])) {
+		SWAP(r1, r0);
+	}
+	ERR_FAIL_COND(0.0 == r0[0]);
+
+	/* eliminate first variable     */
+	m1 = r1[0] / r0[0];
+	m2 = r2[0] / r0[0];
+	m3 = r3[0] / r0[0];
+	s = r0[1];
+	r1[1] -= m1 * s;
+	r2[1] -= m2 * s;
+	r3[1] -= m3 * s;
+	s = r0[2];
+	r1[2] -= m1 * s;
+	r2[2] -= m2 * s;
+	r3[2] -= m3 * s;
+	s = r0[3];
+	r1[3] -= m1 * s;
+	r2[3] -= m2 * s;
+	r3[3] -= m3 * s;
+	s = r0[4];
+	if (s != 0.0) {
+		r1[4] -= m1 * s;
+		r2[4] -= m2 * s;
+		r3[4] -= m3 * s;
+	}
+	s = r0[5];
+	if (s != 0.0) {
+		r1[5] -= m1 * s;
+		r2[5] -= m2 * s;
+		r3[5] -= m3 * s;
+	}
+	s = r0[6];
+	if (s != 0.0) {
+		r1[6] -= m1 * s;
+		r2[6] -= m2 * s;
+		r3[6] -= m3 * s;
+	}
+	s = r0[7];
+	if (s != 0.0) {
+		r1[7] -= m1 * s;
+		r2[7] -= m2 * s;
+		r3[7] -= m3 * s;
 	}
 
-	/* That was most of the work, one final pass of row/column interchange */
-	/* to finish */
-	for (k = 4 - 2; k >= 0; k--) { /* Don't need to work with 1 by 1 corner*/
-		i = pvt_j[k]; /* Rows to swap correspond to pivot COLUMN */
-		if (i != k) { /* If rows are different */
-			for (j = 0; j < 4; j++) {
-				hold = columns[k][j];
-				columns[k][j] = -columns[i][j];
-				columns[i][j] = hold;
-			}
-		}
-
-		j = pvt_i[k]; /* Columns to swap correspond to pivot ROW */
-		if (j != k) { /* If columns are different */
-			for (i = 0; i < 4; i++) {
-				hold = columns[i][k];
-				columns[i][k] = -columns[i][j];
-				columns[i][j] = hold;
-			}
-		}
+	/* choose pivot - or die */
+	if (Math::abs(r3[1]) > Math::abs(r2[1])) {
+		SWAP(r3, r2);
 	}
+	if (Math::abs(r2[1]) > Math::abs(r1[1])) {
+		SWAP(r2, r1);
+	}
+	ERR_FAIL_COND(0.0 == r1[1]);
+
+	/* eliminate second variable */
+	m2 = r2[1] / r1[1];
+	m3 = r3[1] / r1[1];
+	r2[2] -= m2 * r1[2];
+	r3[2] -= m3 * r1[2];
+	r2[3] -= m2 * r1[3];
+	r3[3] -= m3 * r1[3];
+	s = r1[4];
+	if (0.0 != s) {
+		r2[4] -= m2 * s;
+		r3[4] -= m3 * s;
+	}
+	s = r1[5];
+	if (0.0 != s) {
+		r2[5] -= m2 * s;
+		r3[5] -= m3 * s;
+	}
+	s = r1[6];
+	if (0.0 != s) {
+		r2[6] -= m2 * s;
+		r3[6] -= m3 * s;
+	}
+	s = r1[7];
+	if (0.0 != s) {
+		r2[7] -= m2 * s;
+		r3[7] -= m3 * s;
+	}
+
+	/* choose pivot - or die */
+	if (Math::abs(r3[2]) > Math::abs(r2[2])) {
+		SWAP(r3, r2);
+	}
+	ERR_FAIL_COND(0.0 == r2[2]);
+
+	/* eliminate third variable */
+	m3 = r3[2] / r2[2];
+	r3[3] -= m3 * r2[3];
+	r3[4] -= m3 * r2[4];
+	r3[5] -= m3 * r2[5];
+	r3[6] -= m3 * r2[6];
+	r3[7] -= m3 * r2[7];
+
+	/* last check */
+	ERR_FAIL_COND(0.0 == r3[3]);
+
+	s = 1.0 / r3[3]; /* now back substitute row 3 */
+	r3[4] *= s;
+	r3[5] *= s;
+	r3[6] *= s;
+	r3[7] *= s;
+
+	m2 = r2[3]; /* now back substitute row 2 */
+	s = 1.0 / r2[2];
+	r2[4] = s * (r2[4] - r3[4] * m2);
+	r2[5] = s * (r2[5] - r3[5] * m2);
+	r2[6] = s * (r2[6] - r3[6] * m2);
+	r2[7] = s * (r2[7] - r3[7] * m2);
+	m1 = r1[3];
+	r1[4] -= r3[4] * m1;
+	r1[5] -= r3[5] * m1;
+	r1[6] -= r3[6] * m1;
+	r1[7] -= r3[7] * m1;
+	m0 = r0[3];
+	r0[4] -= r3[4] * m0;
+	r0[5] -= r3[5] * m0;
+	r0[6] -= r3[6] * m0;
+	r0[7] -= r3[7] * m0;
+
+	m1 = r1[2]; /* now back substitute row 1 */
+	s = 1.0 / r1[1];
+	r1[4] = s * (r1[4] - r2[4] * m1);
+	r1[5] = s * (r1[5] - r2[5] * m1),
+	r1[6] = s * (r1[6] - r2[6] * m1);
+	r1[7] = s * (r1[7] - r2[7] * m1);
+	m0 = r0[2];
+	r0[4] -= r2[4] * m0;
+	r0[5] -= r2[5] * m0;
+	r0[6] -= r2[6] * m0;
+	r0[7] -= r2[7] * m0;
+
+	m0 = r0[1]; /* now back substitute row 0 */
+	s = 1.0 / r0[0];
+	r0[4] = s * (r0[4] - r1[4] * m0);
+	r0[5] = s * (r0[5] - r1[5] * m0),
+	r0[6] = s * (r0[6] - r1[6] * m0);
+	r0[7] = s * (r0[7] - r1[7] * m0);
+
+	MAT(out, 0, 0) = r0[4];
+	MAT(out, 0, 1) = r0[5];
+	MAT(out, 0, 2) = r0[6];
+	MAT(out, 0, 3) = r0[7];
+	MAT(out, 1, 0) = r1[4];
+	MAT(out, 1, 1) = r1[5];
+	MAT(out, 1, 2) = r1[6];
+	MAT(out, 1, 3) = r1[7];
+	MAT(out, 2, 0) = r2[4];
+	MAT(out, 2, 1) = r2[5];
+	MAT(out, 2, 2) = r2[6];
+	MAT(out, 2, 3) = r2[7];
+	MAT(out, 3, 0) = r3[4];
+	MAT(out, 3, 1) = r3[5];
+	MAT(out, 3, 2) = r3[6];
+	MAT(out, 3, 3) = r3[7];
+
+#undef MAT
+
+	*this = temp;
 }
 
 void Projection::flip_y() {
diff --git a/tests/core/math/test_projection.h b/tests/core/math/test_projection.h
new file mode 100644
index 00000000000..7f41aaee3e7
--- /dev/null
+++ b/tests/core/math/test_projection.h
@@ -0,0 +1,87 @@
+/**************************************************************************/
+/*  test_projection.h                                                     */
+/**************************************************************************/
+/*                         This file is part of:                          */
+/*                             GODOT ENGINE                               */
+/*                        https://godotengine.org                         */
+/**************************************************************************/
+/* Copyright (c) 2014-present Godot Engine contributors (see AUTHORS.md). */
+/* Copyright (c) 2007-2014 Juan Linietsky, Ariel Manzur.                  */
+/*                                                                        */
+/* Permission is hereby granted, free of charge, to any person obtaining  */
+/* a copy of this software and associated documentation files (the        */
+/* "Software"), to deal in the Software without restriction, including    */
+/* without limitation the rights to use, copy, modify, merge, publish,    */
+/* distribute, sublicense, and/or sell copies of the Software, and to     */
+/* permit persons to whom the Software is furnished to do so, subject to  */
+/* the following conditions:                                              */
+/*                                                                        */
+/* The above copyright notice and this permission notice shall be         */
+/* included in all copies or substantial portions of the Software.        */
+/*                                                                        */
+/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,        */
+/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF     */
+/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. */
+/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY   */
+/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,   */
+/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE      */
+/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.                 */
+/**************************************************************************/
+
+#ifndef TEST_PROJECTION_H
+#define TEST_PROJECTION_H
+
+#include "core/math/projection.h"
+
+#include "core/string/print_string.h"
+#include "tests/test_macros.h"
+
+namespace TestProjection {
+
+TEST_CASE("[Projection] Default construct") {
+	Projection p;
+	CHECK(p.columns[0][0] == 1.0);
+	CHECK(p.columns[0][1] == 0.0);
+	CHECK(p.columns[0][2] == 0.0);
+	CHECK(p.columns[0][3] == 0.0);
+
+	CHECK(p.columns[1][0] == 0.0);
+	CHECK(p.columns[1][1] == 1.0);
+	CHECK(p.columns[1][2] == 0.0);
+	CHECK(p.columns[1][3] == 0.0);
+
+	CHECK(p.columns[2][0] == 0.0);
+	CHECK(p.columns[2][1] == 0.0);
+	CHECK(p.columns[2][2] == 1.0);
+	CHECK(p.columns[2][3] == 0.0);
+
+	CHECK(p.columns[3][0] == 0.0);
+	CHECK(p.columns[3][1] == 0.0);
+	CHECK(p.columns[3][2] == 0.0);
+	CHECK(p.columns[3][3] == 1.0);
+}
+
+bool projection_is_equal_approx(const Projection &p_a, const Projection &p_b) {
+	for (int i = 0; i < 4; i++) {
+		for (int j = 0; j < 4; j++) {
+			if (!Math::is_equal_approx(p_a.columns[i][j], p_b.columns[i][j])) {
+				return false;
+			}
+		}
+	}
+	return true;
+}
+
+TEST_CASE("[Projection] Orthogonal projection matrix inversion") {
+	Projection p = Projection::create_orthogonal(-125.0f, 125.0f, -125.0f, 125.0f, 0.01f, 25.0f);
+	CHECK(projection_is_equal_approx(p.inverse() * p, Projection()));
+}
+
+TEST_CASE("[Projection] Perspective projection matrix inversion") {
+	Projection p = Projection::create_perspective(90.0f, 1.77777f, 0.05f, 4000.0f);
+	CHECK(projection_is_equal_approx(p.inverse() * p, Projection()));
+}
+
+} //namespace TestProjection
+
+#endif // TEST_PROJECTION_H
diff --git a/tests/test_main.cpp b/tests/test_main.cpp
index edadc52a163..b46a62899d4 100644
--- a/tests/test_main.cpp
+++ b/tests/test_main.cpp
@@ -61,6 +61,7 @@
 #include "tests/core/math/test_geometry_3d.h"
 #include "tests/core/math/test_math_funcs.h"
 #include "tests/core/math/test_plane.h"
+#include "tests/core/math/test_projection.h"
 #include "tests/core/math/test_quaternion.h"
 #include "tests/core/math/test_random_number_generator.h"
 #include "tests/core/math/test_rect2.h"