求矩阵的秩的时候可以混合使用初等行变换和初等列变换吗？ 矩阵的初等行变换和列变换混用求矩阵的秩
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在线性代数中，一个矩阵A的列秩是A的线性独立的纵列的极大数目。类似地，行秩是A的线性无关的横行的极大数目。即如果把矩阵看成一个个行向量或者列向量，秩就是这些行向量或者列向量的秩，也就是极大无关组中所含向量的个数。
A=(aij)m×n的不为零的子式的最大阶数称为矩阵A的秩，记作rA，或rankA或R(A)。特别规定零矩阵的秩为零。显然rA≤min(m,n) 易得：若A中至少有一个r阶子式不等于零，且在r<min(m,n)时，A中所有的r+1阶子式全为零，则A的秩为r。
由定义直接可得n阶可逆矩阵的秩为n，通常又将可逆矩阵称为满秩矩阵, det(A)≠0；不满秩矩阵就是奇异矩阵，det(A)=0。由行列式的性质知，矩阵A的转置AT的秩与A的秩是一样的，即rank(A)=rank(AT)。
扩展资料：
矩阵的秩的性质：
1、转置后秩不变；
2、r(A)<=min(m,n),A是m*n型矩阵；
3、r(kA)=r(A),k不等于0；
4、r(A)=0 <=> A=0；
5、r(A+B)<=r(A)+r(B)；
6、r(AB)<=min(r(A),r(B))；
7、r(A)+r(B)-n<=r(AB)。
若题目让求一个矩阵阶梯形矩阵和约化的阶梯形矩阵
则只能用初等行变换.
只求矩阵的秩的话, 可以行列变换混用
不过行变换足够用了
若求极大无关组或解线性方程组, 则只能用初等行变换
有列阶梯矩阵这一说, 但大部分教材不提它
行阶梯形:
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