如何解决矩阵的相等性？

一个矩形或正方形的数字或符号数组以行和列的形式组织起来，以表示一个数学对象或其属性之一，在数学中称为矩阵。例如， $\begin{vmatrix}{a}&b&c\\p&q&r\\s&t&u\end{vmatrix}$是一个 3 行 3 列的矩阵。它可以称为 3×3 矩阵，是方阵。另一方面， $\begin{vmatrix}{a}&b&c\\p&q&r\end{vmatrix}$是一个矩形矩阵。

解决矩阵的相等性

当两个或多个矩阵相等时，称为矩阵相等。如果矩阵具有相同的行数和列数以及相同数量的元素，则认为矩阵是相等的。对于前面提到的任何一个条件，矩阵的相等性都不成立。如果矩阵的顺序不相等或至少一对对应的元素不相等，则称这两个矩阵不相等。这个概念与矩形和方形矩阵有关。

下面列出了矩阵等式的三个要求：

• 矩阵 A 和 B 中的行数相同，即 m = p。
• 矩阵 A 和 B 中的列数相同，即 n = q。
• 对于任意 i 和 j，A 和 B 的对应元素相等，即 a _ij = b _ij 。

例子：

Say $\begin{vmatrix}{3}&2&4\\a&6&z\end{vmatrix}=\begin{vmatrix}{3}&2&4\\69&6&420\end{vmatrix}$. Find the values of a and z.

Because the order of the two matrices is equal, matrices are equal if and only if their corresponding elements are likewise equal.

Thus, comparing a and c to the corresponding elements of the other matrix, we have a = 69 and z = 420.

示例问题

问题1：矩阵$\left[\begin{array}{ccc} 2 & 3 & 6 \\ 4 & 5 & 7 \\ 1 & -3 & -12 \end{array}\right]$和$\left[\begin{array}{ccc} -1 & 3 & 6 \\ 4 & 5 & 7 \\ 1 & -3 & -12 \end{array}\right]$平等的？

解决方案：

The given matrices have the same order, which means they have the same number of rows and columns. As a result, the first criterion for matrix equality has been met. The first condition of equality of matrices is fulfilled. Now the second condition, i.e., the corresponding elements must be equal needs to be examined. As can be seen, the element in the first row and first column of the first matrix is 2 in the first matrix and -1 in the second matrix, indicating that not all the elements are equal. 

Hence the two matrices are not equal.

问题 2：如果 A = [a+b 6 8 2x 3b] 和 B = [3 6 8 14 9] 相等，则求 a、b、x 的值。

解决方案：

Since A and B are given to be equal matrices, therefore their corresponding elements are also equal. We have

a + b = 3, 2x = 14, 3b = 9

⇒ x = 24/2 = 12, b = 9/3 = 3

⇒ a + 3 = 3 [From b = 3]

⇒ a = 0

⇒ a = 0, b = 3, x = 7

问题2：如果$\left[\begin{array}{ccc} 3x+4y & x-2y & 6 \\ a+b & -3 & 2a-b \end{array}\right] = \left[\begin{array}{ccc} 2 & 4 & 6 \\ 5 & -3 & -5 \end{array}\right]$ ，求 a、b、x 和 y 的值。

解决方案：

3x + 4y = 2, ⇢ (1)

x – 2y = 4, ⇢  (2)

a + b = 5, ⇢  (3)

2a – b = -5 ⇢  (4)

Solving equations (1) and (2), 

x = 2y + 4 [From (2)]

Substituting the above in (1),

3(2y + 4) + 4y = 2

⇒ 6y + 12 + 4y = 2

⇒ 10y = 2 – 12

⇒ 10y = -10

⇒ y = -1

⇒ x = 2(-1) + 4

= -2 + 4

= 2

Similarly, solving equations (3) and (4), we have a = 0 and b = 5.

问题 4：如果$\left[\begin{array}{ccc}3& x+y\\x-y&5\end{array}\right]=\left[\begin{array}{ccc}3&-7\\2&5\end{array}\right]$ ，求 x 和 y 的值。

Since A and B are given to be equal matrices, therefore their corresponding elements are also equal. We have

x + y = -7

x – y = 2

Adding the two equations, we have:

2x = -5

x = -5/2

So -5/2 -2 = 7

y = 9/2

问题5：矩阵$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 8 & 4 & 6 \\ 4 & 5 & 7 \end{array}\right]$和$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 8 & 5 & 6 \\ 4 & 5 & 7 \end{array}\right]$平等的？

解决方案：

The given matrices have the same order, which means they have the same number of rows and columns. As a result, the first criterion for matrix equality has been met. The first condition of equality of matrices is fulfilled. Now the second condition, i.e., the corresponding elements must be equal needs to be examined. As can be seen, the element in the second row and second column of the first matrix is 4 in the first matrix and 5 in the second matrix, indicating that not all the elements are equal.

Hence the two matrices are not equal.