What is the internals of the mul( float4, matrix4x4) operation for HLSL (version ps_2_0)?

Question

Shaders

There is such a pixel shader that performs a perspective transformation of texture coordinates.

float4x4 matrix_transform;
struct PS_INPUT
{
    float2  Tex0    : TEXCOORD0;
    float4	diffuse	: COLOR;
};
float2 GetCoord(float2 coord_uv)
{
  float4 tex_coord = float4(coord_uv.x, coord_uv.y, 0.0, 1.0);
  tex_coord = mul(tex_coord, matrix_transform);
  return float2(tex_coord.x / tex_coord.w + 0.5, tex_coord.y / tex_coord.w + 0.5);
}
float4 main(PS_INPUT input) : COLOR0
{
  return tex2D(texture0, GetCoord(input.Tex0));
}

It turns out that for the correct execution of the mul operation, before sending the matrix_transform matrix, you have to transpose it.

...
       D3DXMATRIX _matrix;
...
        _shaderPixel.Bind();
        D3DXMatrixTranspose(&_matrix, &_matrix);
  _shaderPixel.setPSParam("matrix_transform", _matrix, 16);
...

Those. position of columns and rows in HLSL and DirectX is mixed up! Why is this happening? Can anyone tell?
Did you find out about such an opportunity yourself only after disassembling the shader code? Couldn't find any information anywhere.
My disassembled code:

// Registers:
//
//   Name             Reg   Size
//   ---------------- ----- ----
//   matrix_transform c0       4
//   texture0         s0       1
//

/*
    ps_2_0
    def c4, 0.5, 0, 0, 1
    dcl t0.xy //текстурные координаты input.Tex0
    dcl_2d s0 //texture0
    mov r0.xy, t0 //r0 -> float4 tex_coord
    mov r0.zw, c4
    dp4 r1.w, r0, c3 //r1.w => tex_coord.w after mul
    rcp r1.x, r1.w		//r1.x => tex_coord.w after mul
    dp4 r1.y, r0, c0  //r1.y -> tex_coord.x after mul
    dp4 r0.x, r0, c1  //r0.x -> tex_coord.y after mul
    mad r0.y, r0.x, r1.x, c4.x  //r0.y => ret.y = tex_coord.y/tex_coord.w + 0.5
    mad r0.x, r1.y, r1.x, c4.x	//r0.x => ret.x = tex_coord.x/tex_coord.w + 0.5
    texld r0, r0, s0
    mov oC0, r0
  
  tex_coord after mul -> tex_coord(r1.y, r0.x, -- , r1.x)
*/