diff --git a/Mathlib.lean b/Mathlib.lean
index 6ee865b67bbd0..5d45c7ac55a8d 100644
--- a/Mathlib.lean
+++ b/Mathlib.lean
@@ -1433,6 +1433,7 @@ import Mathlib.Data.Matrix.CharP
 import Mathlib.Data.Matrix.DMatrix
 import Mathlib.Data.Matrix.DualNumber
 import Mathlib.Data.Matrix.Hadamard
+import Mathlib.Data.Matrix.Invertible
 import Mathlib.Data.Matrix.Kronecker
 import Mathlib.Data.Matrix.Notation
 import Mathlib.Data.Matrix.PEquiv
diff --git a/Mathlib/Data/Matrix/Invertible.lean b/Mathlib/Data/Matrix/Invertible.lean
new file mode 100644
index 0000000000000..48e228706613a
--- /dev/null
+++ b/Mathlib/Data/Matrix/Invertible.lean
@@ -0,0 +1,114 @@
+/-
+Copyright (c) 2023 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+
+! This file was ported from Lean 3 source module data.matrix.invertible
+! leanprover-community/mathlib commit 722b3b152ddd5e0cf21c0a29787c76596cb6b422
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.Invertible
+import Mathlib.Data.Matrix.Basic
+
+/-! # Extra lemmas about invertible matrices
+
+Many of the `Invertible` lemmas are about `*`; this restates them to be about `⬝`.
+
+For lemmas about the matrix inverse in terms of the determinant and adjugate, see `Matrix.inv`
+in `LinearAlgebra/Matrix/NonsingularInverse.lean`.
+-/
+
+
+open scoped Matrix
+
+variable {m n : Type _} {α : Type _}
+
+variable [Fintype n] [DecidableEq n] [Semiring α]
+
+namespace Matrix
+
+/-- A copy of `invOf_mul_self` using `⬝` not `*`. -/
+protected theorem invOf_mul_self (A : Matrix n n α) [Invertible A] : ⅟ A ⬝ A = 1 :=
+  invOf_mul_self A
+#align matrix.inv_of_mul_self Matrix.invOf_mul_self
+
+/-- A copy of `mul_invOf_self` using `⬝` not `*`. -/
+protected theorem mul_invOf_self (A : Matrix n n α) [Invertible A] : A ⬝ ⅟ A = 1 :=
+  mul_invOf_self A
+#align matrix.mul_inv_of_self Matrix.mul_invOf_self
+
+/-- A copy of `invOf_mul_self_assoc` using `⬝` not `*`. -/
+protected theorem invOf_mul_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
+    ⅟ A ⬝ (A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.one_mul]
+#align matrix.inv_of_mul_self_assoc Matrix.invOf_mul_self_assoc
+
+/-- A copy of `mul_invOf_self_assoc` using `⬝` not `*`. -/
+protected theorem mul_invOf_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
+    A ⬝ (⅟ A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.one_mul]
+#align matrix.mul_inv_of_self_assoc Matrix.mul_invOf_self_assoc
+
+/-- A copy of `mul_invOf_mul_self_cancel` using `⬝` not `*`. -/
+protected theorem mul_invOf_mul_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
+    A ⬝ ⅟ B ⬝ B = A := by rw [Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.mul_one]
+#align matrix.mul_inv_of_mul_self_cancel Matrix.mul_invOf_mul_self_cancel
+
+/-- A copy of `mul_mul_invOf_self_cancel` using `⬝` not `*`. -/
+protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
+    A ⬝ B ⬝ ⅟ B = A := by rw [Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.mul_one]
+#align matrix.mul_mul_inv_of_self_cancel Matrix.mul_mul_invOf_self_cancel
+
+/-- A copy of `invertibleMul` using `⬝` not `*`. -/
+@[reducible]
+protected def invertibleMul (A B : Matrix n n α) [Invertible A] [Invertible B] :
+    Invertible (A ⬝ B) :=
+  { invertibleMul _ _ with invOf := ⅟ B ⬝ ⅟ A }
+#align matrix.invertible_mul Matrix.invertibleMul
+
+/-- A copy of `Invertible.mul` using `⬝` not `*`.-/
+@[reducible]
+def _root_.Invertible.matrixMul {A B : Matrix n n α} (_ : Invertible A) (_ : Invertible B) :
+    Invertible (A ⬝ B) :=
+  invertibleMul _ _
+#align invertible.matrix_mul Invertible.matrixMul
+
+protected theorem invOf_mul {A B : Matrix n n α} [Invertible A] [Invertible B]
+    [Invertible (A ⬝ B)] : ⅟ (A ⬝ B) = ⅟ B ⬝ ⅟ A :=
+  invOf_mul _ _
+#align matrix.inv_of_mul Matrix.invOf_mul
+
+/-- A copy of `invertibleOfInvertibleMul` using `⬝` not `*`. -/
+@[reducible]
+protected def invertibleOfInvertibleMul (a b : Matrix n n α) [Invertible a] [Invertible (a ⬝ b)] :
+    Invertible b :=
+  { invertibleOfInvertibleMul a b with invOf := ⅟ (a ⬝ b) ⬝ a }
+#align matrix.invertible_of_invertible_mul Matrix.invertibleOfInvertibleMul
+
+/-- A copy of `invertibleOfMulInvertible` using `⬝` not `*`. -/
+@[reducible]
+protected def invertibleOfMulInvertible (a b : Matrix n n α) [Invertible (a ⬝ b)] [Invertible b] :
+    Invertible a :=
+  { invertibleOfMulInvertible a b with invOf := b ⬝ ⅟ (a ⬝ b) }
+#align matrix.invertible_of_mul_invertible Matrix.invertibleOfMulInvertible
+
+end Matrix
+
+/-- A copy of `Invertible.mulLeft` using `⬝` not `*`. -/
+@[reducible]
+def Invertible.matrixMulLeft {a : Matrix n n α} (_ : Invertible a) (b : Matrix n n α) :
+    Invertible b ≃ Invertible (a ⬝ b) where
+  toFun _ := Matrix.invertibleMul a b
+  invFun _ := Matrix.invertibleOfInvertibleMul a _
+  left_inv _ := Subsingleton.elim _ _
+  right_inv _ := Subsingleton.elim _ _
+#align invertible.matrix_mul_left Invertible.matrixMulLeft
+
+/-- A copy of `Invertible.mulRight` using `⬝` not `*`. -/
+@[reducible]
+def Invertible.matrixMulRight (a : Matrix n n α) {b : Matrix n n α} (_ : Invertible b) :
+    Invertible a ≃ Invertible (a ⬝ b) where
+  toFun _ := Matrix.invertibleMul a b
+  invFun _ := Matrix.invertibleOfMulInvertible _ b
+  left_inv _ := Subsingleton.elim _ _
+  right_inv _ := Subsingleton.elim _ _
+#align invertible.matrix_mul_right Invertible.matrixMulRight
diff --git a/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean b/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
index d60297fca6a3c..5ae6364c8dfc7 100644
--- a/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
+++ b/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Tim Baanen, Lu-Ming Zhang
 
 ! This file was ported from Lean 3 source module linear_algebra.matrix.nonsingular_inverse
-! leanprover-community/mathlib commit a07a7ae98384cd6485d7825e161e528ba78ef3bc
+! leanprover-community/mathlib commit 722b3b152ddd5e0cf21c0a29787c76596cb6b422
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathlib.Data.Matrix.Invertible
 import Mathlib.LinearAlgebra.Matrix.Adjugate
 
 /-!
@@ -67,36 +68,6 @@ section Invertible
 
 variable [Fintype n] [DecidableEq n] [CommRing α]
 
-/-- A copy of `invOf_mul_self` using `⬝` not `*`. -/
-protected theorem invOf_mul_self (A : Matrix n n α) [Invertible A] : ⅟ A ⬝ A = 1 :=
-  invOf_mul_self A
-#align matrix.inv_of_mul_self Matrix.invOf_mul_self
-
-/-- A copy of `mul_invOf_self` using `⬝` not `*`. -/
-protected theorem mul_invOf_self (A : Matrix n n α) [Invertible A] : A ⬝ ⅟ A = 1 :=
-  mul_invOf_self A
-#align matrix.mul_inv_of_self Matrix.mul_invOf_self
-
-/-- A copy of `invOf_mul_self_assoc` using `⬝` not `*`. -/
-protected theorem invOf_mul_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
-    ⅟ A ⬝ (A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.one_mul]
-#align matrix.inv_of_mul_self_assoc Matrix.invOf_mul_self_assoc
-
-/-- A copy of `mul_invOf_self_assoc` using `⬝` not `*`. -/
-protected theorem mul_invOf_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
-    A ⬝ (⅟ A ⬝ B) = B := by rw [← Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.one_mul]
-#align matrix.mul_inv_of_self_assoc Matrix.mul_invOf_self_assoc
-
-/-- A copy of `mul_invOf_mul_self_cancel` using `⬝` not `*`. -/
-protected theorem mul_invOf_mul_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
-    A ⬝ ⅟ B ⬝ B = A := by rw [Matrix.mul_assoc, Matrix.invOf_mul_self, Matrix.mul_one]
-#align matrix.mul_inv_of_mul_self_cancel Matrix.mul_invOf_mul_self_cancel
-
-/-- A copy of `mul_mul_invOf_self_cancel` using `⬝` not `*`. -/
-protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
-    A ⬝ B ⬝ ⅟ B = A := by rw [Matrix.mul_assoc, Matrix.mul_invOf_self, Matrix.mul_one]
-#align matrix.mul_mul_inv_of_self_cancel Matrix.mul_mul_invOf_self_cancel
-
 variable (A : Matrix n n α) (B : Matrix n n α)
 
 /-- If `A.det` has a constructive inverse, produce one for `A`. -/