Matrix Inverse Explained
Before heading to the matrix inverse properties, it is crucial to first understand the meaning and mechanism of the matrix as well the inverse of a matrix. That said, Matrices are robust mathematical tools that can be used in making computer games and all the exciting stuff that appears on the computer screen. Besides, Computer gaming and game graphics, it also has a huge array of applications that ranges from:-
Huge data visualization,
Structuring buildings resistant to earthquakes or natural calamities
Measuring infant mortality and beyond
Definition of Matrix and the Inverse of a Matrix
Matrices, although are complex to use manually, are very useful when incorporated through computers and. Matrices are capable of solving cumbersome problems very easily and quickly.
There are ways to add, subtract and multiply matrices, but there is no concept of division of a matrix operation. And this is where the inverse of a matrix comes into action. The nearest we can reach to division of a matrix is—multiplying by its inverse.
For all you know from functions with integer numbers that dividing by a number gets you the same answer as multiplying by its reciprocal. 20/10= 10 x (1/10) = 2. The same holds truth for the matrix inverses – until that matrix has an inverse.
Inverse Matrix Properties
There are a couple of inverse properties to take into account when talking about the inverse of a matrix. First, if multiplying a matrix by its inverse, the sequence does not matter. This is largely atypical for matrix functions because XZ barely equals ZX for the majority of matrices. Second, the inverse of a matrix may not even exist. In case the determinant of a matrix is zero, you just can NOT divide by that!
There Are Basically 3 Other Properties Of The Inverse As Below:-
1. If X is a square matrix and Z is the inverse of X, then X is the inverse
Of Z, since XZ = I = ZX. Then we acquire the identity inverse:
(X−1)−1 = X
2. Observe that Z−1X−1XZ = B−1
IZ = I = XZZ−1X−1
. Then:
(XZ)−1 = Z−1X−1
Then in a similar manner as transpose, taking the inverse of a product reverse the sequence of the product.
3. Finally, bear in mind that (XZ)
T = ZT XT
. Since I
T = I, then (X−1X)
T =XT (X−1) T = I.
Likewise, (XX−1) T = (X−1) T XT = I.
Then:
(X−1) T = (XT)−1
Therefore, we could even M−1V write X−T for the inverse transpose of X
(Image to be added soon)
Need of an Inverse
Since with matrices there is no such operation of division, thus we don't divide by a matrix. However, we can multiply by an inverse, which accomplishes the same purpose.
Solved Example
Just Imagine we cannot divide by numbers and someone asks "How do I distribute 20 mangoes amongst 4 people?" But we can still use the reciprocal of 4 (which is 0.25), so we answer:
Likewise the same theory can be applied with matrices:
Say we want to find matrix O, and we know matrix M and N:
OM = BN
It would be good to divide both sides by M (to obtain O=N/M), but recall we can't divide.
But, what if we multiply both sides by M-1 ?
OMM-1 = NM-1
And we notice that MM-1 = I, so:
OI = NM-1
Here, we can eliminate I (for the same logic we can eliminate "1" from 1o = mn for numbers):
O = NM-1
And we get our answer (supposing we can calculate A-1)
MN is almost never equal to NM.
Calculate the Inverse of a 2x2 Matrix Operations
To obtain the inverse of a 2x2 matrix, you will require following a few steps:
Swap the numbers in (row 1 & column 1) and (row 2 & column 2)
Give opposite signs to the numbers in (row 1 & column 2) and (row 2 & column 1)
Now, finally divide by the determinant of the native matrix
Fun Facts
The Inverse May Not Exist
With matrices the sequence of multiplication matters
Firstly, to get an inverse the matrix should be "square" of same number of rows and columns
The determinant cannot be zero (or we finish as dividing by 0)
Only computer can find inverses to most real systems of equations that are very big (up to 100 by 100 is)
FAQs on Properties of Matrices Inverse
1. What are the essential conditions for a matrix to have an inverse?
For a matrix to be invertible, it must satisfy two fundamental conditions as per the CBSE Class 12 syllabus for 2025-26. First, the matrix must be a square matrix, meaning it has the same number of rows and columns (e.g., 2x2, 3x3). Second, the determinant of the matrix must be non-zero. A matrix with a zero determinant is called a singular matrix, and its inverse does not exist.
2. What are the key properties of the inverse of a matrix?
The inverse of a matrix has several important properties that are crucial for solving problems. If A and B are invertible matrices, the main properties are:
- Inverse of an Inverse: The inverse of an inverse matrix is the original matrix itself. (A⁻¹)⁻¹ = A.
- Reversal Law for Inverses: The inverse of the product of two matrices is the product of their inverses in the reverse order. (AB)⁻¹ = B⁻¹A⁻¹.
- Inverse of a Transpose: The inverse of the transpose of a matrix is equal to the transpose of its inverse. (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
- Scalar Multiplication: The inverse of a matrix multiplied by a non-zero scalar 'k' is the inverse of the matrix multiplied by the reciprocal of the scalar. (kA)⁻¹ = (1/k)A⁻¹.
3. How is the inverse of a matrix used to solve a system of linear equations?
The inverse of a matrix provides a powerful method for solving a system of linear equations. A system of equations, like a₁x + b₁y = c₁ and a₂x + b₂y = c₂, can be represented in matrix form as AX = B. Here, A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. To solve for the variables in matrix X, we can pre-multiply both sides by the inverse of A (A⁻¹), which gives: X = A⁻¹B. This method is analogous to dividing by a coefficient in simple algebra and is a key application of matrix inverses.
4. What is the general method to find the inverse of a 3x3 matrix using its adjoint?
To find the inverse of a 3x3 matrix (A) using the adjoint method, you follow these steps:
- Step 1: Calculate the determinant of A, denoted as det(A) or |A|. If det(A) is zero, the inverse does not exist.
- Step 2: Find the matrix of cofactors for each element of A.
- Step 3: Determine the adjoint of A, denoted as adj(A), by taking the transpose of the cofactor matrix.
- Step 4: Use the formula to calculate the inverse: A⁻¹ = (1/det(A)) * adj(A).
5. Why is the inverse of a product of matrices (AB)⁻¹ equal to B⁻¹A⁻¹ and not A⁻¹B⁻¹?
This is known as the reversal law of inverses and is a direct consequence of matrix multiplication not being commutative (i.e., AB ≠ BA). To prove this, we must show that multiplying (AB) by (B⁻¹A⁻¹) results in the identity matrix (I).
Consider: (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹.
Since BB⁻¹ = I (the identity matrix), the expression becomes A(I)A⁻¹ = AA⁻¹.
Finally, since AA⁻¹ = I, we have proven that (B⁻¹A⁻¹) is the correct inverse of (AB). The reverse order is necessary to ensure the inner matrices cancel out to form the identity matrix.
6. Is the inverse of a sum of two matrices, (A+B)⁻¹, equal to the sum of their inverses, A⁻¹ + B⁻¹?
No, this is a common misconception. In general, (A+B)⁻¹ ≠ A⁻¹ + B⁻¹. Unlike scalar algebra, matrix operations do not distribute over addition in this manner. The relationship between matrices, their sums, and their inverses is not linear. There is no simple formula for the inverse of a sum of matrices; it must be calculated by first finding the sum matrix (A+B) and then finding its inverse, provided the sum matrix is invertible.
7. What is the relationship between the determinant of a matrix and the determinant of its inverse?
There is a direct reciprocal relationship between the determinant of a matrix and the determinant of its inverse. If A is an invertible matrix, then the determinant of its inverse, det(A⁻¹), is equal to the reciprocal of the determinant of the original matrix, 1/det(A). This can be derived from the property det(AB) = det(A)det(B). Since AA⁻¹ = I, we have det(AA⁻¹) = det(I), which means det(A)det(A⁻¹) = 1. This property is useful for quickly finding the determinant of an inverse without calculating the entire inverse matrix.
8. What is an identity matrix and why is it important for the concept of matrix inverses?
An identity matrix (I) is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts as the multiplicative identity in matrix algebra, similar to the number 1 in scalar multiplication. Its importance for matrix inverses is definitional: the inverse of a matrix A is another matrix, A⁻¹, such that their product is the identity matrix. That is, AA⁻¹ = A⁻¹A = I. Without the concept of an identity matrix, the definition and verification of a matrix inverse would not be possible.
