diff --git a/stdlib/LinearAlgebra/src/bidiag.jl b/stdlib/LinearAlgebra/src/bidiag.jl
index f8cc3ceadcfad..78c79b6fcefac 100644
--- a/stdlib/LinearAlgebra/src/bidiag.jl
+++ b/stdlib/LinearAlgebra/src/bidiag.jl
@@ -426,6 +426,7 @@ end
 
 const BandedMatrix = Union{Bidiagonal,Diagonal,Tridiagonal,SymTridiagonal} # or BiDiTriSym
 const BiTriSym = Union{Bidiagonal,Tridiagonal,SymTridiagonal}
+const TriSym = Union{Tridiagonal,SymTridiagonal}
 const BiTri = Union{Bidiagonal,Tridiagonal}
 @inline mul!(C::AbstractVector, A::BandedMatrix, B::AbstractVector, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
 @inline mul!(C::AbstractMatrix, A::BandedMatrix, B::AbstractVector, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
@@ -433,6 +434,9 @@ const BiTri = Union{Bidiagonal,Tridiagonal}
 @inline mul!(C::AbstractMatrix, A::AbstractMatrix, B::BandedMatrix, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
 @inline mul!(C::AbstractMatrix, A::BandedMatrix, B::BandedMatrix, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
 
+lmul!(A::Bidiagonal, B::AbstractVecOrMat) = @inline _mul!(B, A, B, MulAddMul())
+rmul!(B::AbstractMatrix, A::Bidiagonal) = @inline _mul!(B, B, A, MulAddMul())
+
 function check_A_mul_B!_sizes(C, A, B)
     mA, nA = size(A)
     mB, nB = size(B)
@@ -460,7 +464,11 @@ function _diag(A::Bidiagonal, k)
     end
 end
 
-function _mul!(C::AbstractMatrix, A::BiTriSym, B::BiTriSym, _add::MulAddMul = MulAddMul())
+_mul!(C::AbstractMatrix, A::BiTriSym, B::TriSym, _add::MulAddMul = MulAddMul()) =
+    _bibimul!(C, A, B, _add)
+_mul!(C::AbstractMatrix, A::BiTriSym, B::Bidiagonal, _add::MulAddMul = MulAddMul()) =
+    _bibimul!(C, A, B, _add)
+function _bibimul!(C, A, B, _add)
     check_A_mul_B!_sizes(C, A, B)
     n = size(A,1)
     n <= 3 && return mul!(C, Array(A), Array(B), _add.alpha, _add.beta)
@@ -583,7 +591,7 @@ function _mul!(C::AbstractVecOrMat, A::BiTriSym, B::AbstractVecOrMat, _add::MulA
     C
 end
 
-function _mul!(C::AbstractMatrix, A::AbstractMatrix, B::BiTriSym, _add::MulAddMul = MulAddMul())
+function _mul!(C::AbstractMatrix, A::AbstractMatrix, B::TriSym, _add::MulAddMul = MulAddMul())
     require_one_based_indexing(C, A)
     check_A_mul_B!_sizes(C, A, B)
     iszero(_add.alpha) && return _rmul_or_fill!(C, _add.beta)
@@ -618,7 +626,37 @@ function _mul!(C::AbstractMatrix, A::AbstractMatrix, B::BiTriSym, _add::MulAddMu
     C
 end
 
-function _mul!(C::AbstractMatrix, A::Diagonal, B::BiTriSym, _add::MulAddMul = MulAddMul())
+function _mul!(C::AbstractMatrix, A::AbstractMatrix, B::Bidiagonal, _add::MulAddMul = MulAddMul())
+    require_one_based_indexing(C, A)
+    check_A_mul_B!_sizes(C, A, B)
+    iszero(_add.alpha) && return _rmul_or_fill!(C, _add.beta)
+    if size(A, 1) <= 3 || size(B, 2) <= 1
+        return mul!(C, Array(A), Array(B), _add.alpha, _add.beta)
+    end
+    m, n = size(A)
+    @inbounds if B.uplo == 'U'
+        for i in 1:m
+            for j in n:-1:2
+                _modify!(_add, A[i,j] * B.dv[j] + A[i,j-1] * B.ev[j-1], C, (i, j))
+            end
+            _modify!(_add, A[i,1] * B.dv[1], C, (i, 1))
+        end
+    else # uplo == 'L'
+        for i in 1:m
+            for j in 1:n-1
+                _modify!(_add, A[i,j] * B.dv[j] + A[i,j+1] * B.ev[j], C, (i, j))
+            end
+            _modify!(_add, A[i,n] * B.dv[n], C, (i, n))
+        end
+    end
+    C
+end
+
+_mul!(C::AbstractMatrix, A::Diagonal, B::Bidiagonal, _add::MulAddMul = MulAddMul()) =
+    _dibimul!(C, A, B, _add)
+_mul!(C::AbstractMatrix, A::Diagonal, B::TriSym, _add::MulAddMul = MulAddMul()) =
+    _dibimul!(C, A, B, _add)
+function _dibimul!(C, A, B, _add)
     require_one_based_indexing(C)
     check_A_mul_B!_sizes(C, A, B)
     n = size(A,1)
diff --git a/stdlib/LinearAlgebra/test/bidiag.jl b/stdlib/LinearAlgebra/test/bidiag.jl
index a3e5a2f437e93..2fce781e30ab1 100644
--- a/stdlib/LinearAlgebra/test/bidiag.jl
+++ b/stdlib/LinearAlgebra/test/bidiag.jl
@@ -439,6 +439,9 @@ Random.seed!(1)
                 for op in (+, -, *)
                     @test Array(op(T, T2)) ≈ op(Tfull, Tfull2)
                 end
+                A = kron(T.dv, T.dv')
+                @test T * A ≈ lmul!(T, copy(A))
+                @test A * T ≈ rmul!(copy(A), T)
             end
             # test pass-through of mul! for SymTridiagonal*Bidiagonal
             TriSym = SymTridiagonal(T.dv, T.ev)
@@ -446,7 +449,8 @@ Random.seed!(1)
             # test pass-through of mul! for AbstractTriangular*Bidiagonal
             Tri = UpperTriangular(diagm(1 => T.ev))
             Dia = Diagonal(T.dv)
-            @test Array(Tri*T) ≈ Array(Tri)*Array(T)
+            @test Array(Tri*T) ≈ Array(Tri)*Array(T) ≈ rmul!(copy(Tri), T)
+            @test Array(T*Tri) ≈ Array(T)*Array(Tri) ≈ lmul!(T, copy(Tri))
             # test mul! itself for these types
             for AA in (Tri, Dia)
                 for f in (identity, transpose, adjoint)
@@ -459,8 +463,10 @@ Random.seed!(1)
             for f in (identity, transpose, adjoint)
                 C = relty == Int ? rand(float(elty), n, n) : rand(elty, n, n)
                 B = rand(elty, n, n)
-                D = copy(C) + 2.0 * Array(T*f(B))
-                mul!(C, T, f(B), 2.0, 1.0) ≈ D
+                D = C + 2.0 * Array(T*f(B))
+                @test mul!(C, T, f(B), 2.0, 1.0) ≈ D
+                @test lmul!(T, copy(f(B))) ≈ T * f(B)
+                @test rmul!(copy(f(B)), T) ≈ f(B) * T
             end
 
             # Issue #31870