Problem 1.1: Matrix (100pts)

Problem

Implement the Matrix class that inherits from the Semiring base class.

The Matrix class represents 3x3 matrices and supports matrix operations. We use a nested list (list of lists) to store the elements of the matrix, where elements[i][j] is the element at row i and column j. For example, the list [[1, 2, 3], [4, 5, 6], [7, 8, 9]] represents the matrix

\[\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}\]

实现继承自 Semiring 基类的 Matrix 类。

Matrix 类表示 3x3 矩阵并支持矩阵运算。我们使用嵌套列表（列表的列表）来存储矩阵的元素，其中 elements[i][j] 是第 i 行第 j 列的元素。例如，列表 [[1, 2, 3], [4, 5, 6], [7, 8, 9]] 表示矩阵

\[\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}\]

The definition of addition and multiplication for matrices is as follows:

Addition: The sum of two matrices is obtained by adding their corresponding elements. That is, if $ A $ and $ B $ are two matrices, then their sum $ C = A + B $ is defined by $ C_{i, j} = A_{i, j} + B_{i, j} $.

Accordingly, the zero element (additive identity) is the matrix with all elements equal to 0, and the additive inverse (negation) of a matrix is obtained by negating each of its elements. * Multiplication: The product of two matrices is obtained by taking the dot product of rows and columns. That is, if $ A $ and $ B $ are two matrices, then their product $ C = A \times B $ is defined by $$ C_{i, j} = \sum_{k=0}^{n} A_{i, k} \times B_{k, j} $$ where $ n $ is $ 3 $ for the 3x3 matrices in this problem.

Accordingly, the one element (multiplicative identity) is the identity matrix, which has 1s on the diagonal and 0s elsewhere.

\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]

Here is the skeleton of the Matrix class you need to implement:

矩阵的加法和乘法的定义如下：

加法：两个矩阵的和是通过将它们对应的元素相加得到的。也就是说，如果 $ A $ 和 $ B $ 是两个矩阵，那么它们的和 $ C = A + B $ 定义为 $ C_{i, j} = A_{i, j} + B_{i, j} $。

因此，零元（加法单位元）是所有元素都等于 0 的矩阵，而一个矩阵的加法逆元（负元）是通过将其每个元素取负得到的。 * 乘法：两个矩阵的乘积是通过取行和列的点积得到的。也就是说，如果 $ A $ 和 $ B $ 是两个矩阵，那么它们的乘积 $ C = A \times B $ 定义为 $$ C_{i, j} = \sum_{k=0}^{n} A_{i, k} \times B_{k, j} $$ 其中在本问题中 $ n $ 对于 3x3 矩阵来说是 $ 3 $。

因此，幺元（乘法单位元）是单位矩阵，它的对角线上是 1，其他地方是 0。

\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]

下面是您需要实现的 Matrix 类的骨架：

class Matrix(Semiring):
    """A class representing 3x3 matrices."""

    def __init__(self, elements):
        """Initializes a 3x3 matrix with the given list of lists.
        >>> Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
        [[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]]
        """
        self.elements = elements

    def add(self, other):
        """Returns a new Matrix representing the sum of self and other.
        >>> m1 = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
        >>> m2 = Matrix([[9, 8, 7], [6, 5, 4], [3, 2, 1]])
        >>> m1.add(m2)
        [[10, 10, 10],
         [10, 10, 10],
         [10, 10, 10]]
        """
        """YOUR CODE HERE"""

    def mult(self, other):
        """Returns a new Matrix representing the product of self and other.
        >>> m1 = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
        >>> m2 = Matrix([[9, 8, 7], [6, 5, 4], [3, 2, 1]])
        >>> m1.mult(m2)
        [[30, 24, 18],
         [84, 69, 54],
         [138, 114, 90]]
        """
        """YOUR CODE HERE"""

    def negative(self):
        """Returns a new Matrix representing the additive inverse of self.
        >>> m = Matrix([[1, -2, 3], [-4, 5, -6], [7, -8, 9]])
        >>> m.negative()
        [[-1, 2, -3],
         [4, -5, 6],
         [-7, 8, -9]]
        """
        """YOUR CODE HERE"""

    def zero(self):
        """Returns the zero matrix."""
        """YOUR CODE HERE"""

    def one(self):
        """Returns the identity matrix."""
        """YOUR CODE HERE"""

    def __repr__(self):
        """Returns the string representation of the matrix.
        You are not supposed to understand this code now.
        """
        rows = [str(row) for row in self.elements]
        return "[{}]".format(",\n ".join(rows))

Problem 1.1: Matrix (100pts)

Problem

Hints