diff --git a/sycl/include/sycl/ext/oneapi/matrix/matrix-jit.hpp b/sycl/include/sycl/ext/oneapi/matrix/matrix-jit.hpp
index e9e03d3b894cb..81e61b1198b96 100644
--- a/sycl/include/sycl/ext/oneapi/matrix/matrix-jit.hpp
+++ b/sycl/include/sycl/ext/oneapi/matrix/matrix-jit.hpp
@@ -238,6 +238,16 @@ class wi_element {
                         PI_INVALID_DEVICE);
 #endif // __SYCL_DEVICE_ONLY__
   }
+
+  explicit operator bool() {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(M.spvm, idx) != static_cast<T>(0);
+#else
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
   wi_element &operator=(const T &rhs) {
 #ifdef __SYCL_DEVICE_ONLY__
     M.spvm = __spirv_VectorInsertDynamic(M.spvm, rhs, idx);
@@ -248,6 +258,80 @@ class wi_element {
                         PI_INVALID_DEVICE);
 #endif // __SYCL_DEVICE_ONLY__
   }
+
+  wi_element &
+  operator=(const wi_element<T, NumRows, NumCols, Layout, Group> &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    M.spvm = __spirv_VectorInsertDynamic(
+        M.spvm, __spirv_VectorExtractDynamic(rhs.M.spvm, rhs.idx), idx);
+    return *this;
+#else
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend T operator+(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+                     const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) + rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  wi_element &operator+=(const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    M.spvm = __spirv_VectorInsertDynamic(
+        M.spvm, __spirv_VectorExtractDynamic(M.spvm, idx) + rhs, idx);
+    return *this;
+#else
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend T operator-(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+                     const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) - rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  wi_element &operator-=(const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    M.spvm = __spirv_VectorInsertDynamic(
+        M.spvm, __spirv_VectorExtractDynamic(M.spvm, idx) - rhs, idx);
+    return *this;
+#else
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend T operator*(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+                     const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) * rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
   wi_element &operator*=(const T &rhs) {
 #ifdef __SYCL_DEVICE_ONLY__
     M.spvm = __spirv_VectorInsertDynamic(
@@ -259,7 +343,108 @@ class wi_element {
                         PI_INVALID_DEVICE);
 #endif // __SYCL_DEVICE_ONLY__
   }
-  // TODO: add other arithmetic operators
+
+  friend T operator/(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+                     const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) / rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  wi_element &operator/=(const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    M.spvm = __spirv_VectorInsertDynamic(
+        M.spvm, __spirv_VectorExtractDynamic(M.spvm, idx) / rhs, idx);
+    return *this;
+#else
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator<(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+            const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) < rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator<=(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+             const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) <= rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator>(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+            const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) > rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator>=(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+             const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) >= rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator==(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+             const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) == rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
+
+  friend bool
+  operator!=(const wi_element<T, NumRows, NumCols, Layout, Group> &lhs,
+             const T &rhs) {
+#ifdef __SYCL_DEVICE_ONLY__
+    return __spirv_VectorExtractDynamic(lhs.M.spvm, lhs.idx) != rhs;
+#else
+    (void)lhs;
+    (void)rhs;
+    throw runtime_error("joint matrix is not supported on host device.",
+                        PI_INVALID_DEVICE);
+#endif // __SYCL_DEVICE_ONLY__
+  }
 };
 
 template <typename T, size_t NumRows, size_t NumCols, matrix_layout Layout,