📚 Inverse Matrices | 逆矩阵
An inverse matrix is the matrix equivalent of a reciprocal. Just as multiplying a number by its reciprocal gives 1, multiplying a matrix by its inverse gives the identity matrix I.
逆矩阵相当于矩阵中的“倒数”。正如一个数乘以它的倒数得到 1，一个矩阵乘以它的逆矩阵会得到单位矩阵 I。
1. What is an inverse matrix? | 什么是逆矩阵
ntity matrix. Only square matrices can have inverses, and not every square matrix has one.
对于方阵 A，它的逆矩阵记作 A⁻¹。它必须满足 AA⁻¹ = A⁻¹A = I，其中 I 是单位矩阵。只有方阵才可能有逆矩阵，而且并非每个方阵都有逆矩阵。
2. The 2 × 2 inverse formula | 2×2 逆矩阵公式
he inverse is given by swapping a and d, changing the signs of b and c, and dividing by the determinant ad − bc.
对于一般的 2×2 矩阵 A = ( a b ; c d )，求逆矩阵的方法是：交换 a 和 d，将 b 和 c 变号，再除以行列式 ad − bc。
A⁻¹ = 1 / (ad − bc) × ( d −b ; −c a )
Here ad − bc is the determinant of A. If this value is zero, the inverse does not exist.
这里的 ad − bc 是 A 的行列式。如果这个值为零，则逆矩阵不存在。
3. Determinant and singular matrices | 行列式与奇异矩阵
is non-singular and has a unique inverse. If det A = 0, A is singular and has no inverse.
2×2 矩阵 A = ( a b ; c d ) 的行列式是 det A = ad − bc。如果 det A ≠ 0，A 是非奇异矩阵，有唯一的逆矩阵。如果 det A = 0，A 是奇异矩阵，没有逆矩阵。
4. Worked example: finding a 2 × 2 inverse | 求 2×2 逆矩阵示例
Find the inverse of A = ( 4 7 ; 2 6 ). First compute det A = 4 × 6 − 7 × 2 = 24 − 14 = 10.
求 A = ( 4 7 ; 2 6 ) 的逆矩阵。先计算 det A = 4 × 6 − 7 × 2 = 24 − 14 = 10。
= 10, A is non-singular. Using the formula, swap 4 and 6, and change the signs of 7 and 2.
因为 det A = 10，A 是非奇异的。根据公式，交换 4 和 6，并将 7 和 2 变号。
A⁻¹ = 1 / 10 × ( 6 −7 ; −2 4 ) = ( 0.6 −0.7 ; −0.2 0.4 )
You can leave the answer as fractions: A⁻¹ = ( 3/5 −7/10 ; −1/5 2/5 ).
答案也可以用分数表示：A⁻¹ = ( 3/5 −7/10 ; −1/5 2/5 )。
5. Verifying an inverse | 验证逆矩阵
by its inverse in both orders. The result should be the identity matrix I = ( 1 0 ; 0 1 ).
检验逆矩阵的方法是将原矩阵与其逆矩阵按两种顺序相乘，结果都应该是单位矩阵 I = ( 1 0 ; 0 1 )。
For the example above, compute A A⁻¹. The product simplifies to ( 1 0 ; 0 1 ), confirming the answer.
对于上面的例子，计算 A A⁻¹，结果会化简为 ( 1 0 ; 0 1 )，从而验证答案正确。
6. Properties of inverse matrices | 逆矩阵的性质
Inverse matrices obey several useful rules. These are important when simplifying matrix expressions in exam questions.
逆矩阵有几个常用的运算性质。这些性质在考试中化简矩阵表达式时非常重要。
- (A⁻¹)⁻¹ = A — the inverse of an inverse is the original matrix.
- (kA)⁻¹ = 1/k × A⁻¹ for a non-zero scalar k.
- (AB)⁻¹ = B⁻¹ A⁻¹ — note the reversed order.
- (Aᵀ)⁻¹ = (A⁻¹)ᵀ — the inverse of a transpose equals the transpose of the inverse.
中文配对：逆矩阵的逆是原矩阵；数乘矩阵的逆等于该数的倒数乘以逆矩阵；乘积的逆等于逆矩阵按相反顺序相乘；转置的逆等于逆的转置。
中文配对：逆矩阵的逆是原矩阵；数乘矩阵的逆等于该数的倒数乘以逆矩阵；乘积的逆等于逆矩阵按相反顺序相乘；转置的逆等于逆的转置。
7. Solving matrix equations using inverses | 用逆矩阵解矩阵方程
iply both sides by A⁻¹ to get X = A⁻¹ B. If XA = B, post-multiply by A⁻¹ to get X = B A⁻¹.
如果 AX = B 且 A 可逆，两边左乘 A⁻¹ 得到 X = A⁻¹ B。如果 XA = B，则两边右乘 A⁻¹ 得到 X = B A⁻¹。
Matrix multiplication is not commutative, so the side on which you multiply matters. Always write A⁻¹ on the same side of both sides of the equation.
矩阵乘法不满足交换律，所以乘以逆矩阵的位置很重要。一定要在等号两边相同的侧乘以 A⁻¹。
8. Solving simultaneous linear equations | 解线性方程组
tion. For example, 2x + 3y = 5 and 4x + y = 11 become ( 2 3 ; 4 1 )( x ; y ) = ( 5 ; 11 ).
二元一次方程组可以写成矩阵方程。例如 2x + 3y = 5 和 4x + y = 11 可以写成 ( 2 3 ; 4 1 )( x ; y ) = ( 5 ; 11 )。
Find the inverse of the coefficient matrix, then pre-multiply the constant vector by this inverse to obtain x and y.
先求系数矩阵的逆矩阵，然后用这个逆矩阵左乘常数向量，就能得到 x 和 y。
9. Inverse of a product | 乘积的逆
For two invertible matrices A and B, the inverse of AB is B⁻¹ A⁻¹. This is a very common trick in A-Level questions.
对于两个可逆矩阵 A 和 B，AB 的逆矩阵是 B⁻¹ A⁻¹。这是 A-Level 题目中非常常见的考点。
the inverse of a product. The same rule extends to three matrices: (ABC)⁻¹ = C⁻¹ B⁻¹ A⁻¹.
求乘积的逆时一定要交换顺序。这一规则可以推广到三个矩阵：(ABC)⁻¹ = C⁻¹ B⁻¹ A⁻¹。
10. Inverse of 3 × 3 matrices: AQA Further Maths overview | 3×3 逆矩阵概述
he formula is A⁻¹ = 1 / det(A) × adj(A), where adj(A) is the adjugate matrix of cofactors.
在 AQA 进阶数学中，你可能还需要求 3×3 矩阵的逆。公式是 A⁻¹ = 1 / det(A) × adj(A)，其中 adj(A) 是由代数余子式组成的伴随矩阵。
ding along a row or column. If det A = 0, the 3 × 3 matrix is singular and has no inverse.
对于 3×3 矩阵，可以先按某一行或某一列展开求行列式。如果 det A = 0，这个 3×3 矩阵就是奇异的，没有逆矩阵。
11. Common mistakes | 常见错误
A common error is forgetting to divide the adjugate matrix by the determinant. The formula is only complete with the factor 1 / det A.
常见错误是忘记将伴随矩阵除以行列式。完整的公式必须包含因子 1 / det A。
ces in the wrong order when dealing with products. Remember (AB)⁻¹ = B⁻¹ A⁻¹, not A⁻¹ B⁻¹.
另一个错误是在处理乘积时把逆矩阵的相乘顺序写反。记住 (AB)⁻¹ = B⁻¹ A⁻¹，而不是 A⁻¹ B⁻¹。
Students also sometimes attempt to invert a singular matrix. Always check that the determinant is non-zero first.
学生有时会尝试对奇异矩阵求逆。一定要先检查行列式是否非零。
12. Exam tips | 考试技巧
on-singular. Showing det A = ad − bc earns method marks even if later arithmetic is wrong.
一定要先写出行列式，并明确说明矩阵是奇异的还是非奇异的。写出 det A = ad − bc 可以在后续计算出错时仍然拿到方法分。
Check your inverse by multiplying A A⁻¹ to see if you get I. This quick check catches sign and division errors.
通过计算 A A⁻¹ 来检验是否得到 I。这种快速检验能发现符号和除法错误。
For matrix equations, write the inverse on the correct side and keep your working neat. A clear layout helps you avoid order mistakes.
解矩阵方程时，要把逆矩阵写在正确的一侧，并保持书写工整。清晰的步骤能帮助你避免顺序错误。
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