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Magazine Chapter 2: Problem 14
Find a left-inverse and/or a right-inverse (when they exist) for $$ A=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right] \quad \text { and } \quad M=\left[\begin{array}{ll} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{array}\right] \quad \text { and } \quad T=\left[\begin{array}{ll} a & b \\ 0 & a \end{array}\right] $$
Short Answer
Expert verified
A has a right-inverse; M has a left-inverse; T has an inverse if \(a \neq 0\).
Step by step solution
01
Understanding Matrix A
Matrix \(A\) is a 2x3 matrix. To find a left-inverse, \(A\) needs to be left-invertible, which happens if its rank is equal to the number of columns (3 in this case). A 2x3 matrix can never have its rank equal the number of columns, so a left-inverse does not exist for \(A\). For a right-inverse, \(A\) needs to be full rank 2 (equal to the number of rows). Since the two rows are linearly independent, \(A\) has full row rank and a right-inverse exists.
02
Finding a Right-Inverse of A
To find a right-inverse of \(A\), we need a 3x2 matrix \(B\) such that \(AB=I_2\). The strategy is to solve the matrix equation \(\begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} x_{11} & x_{12} \ x_{21} & x_{22} \ x_{31} & x_{32} \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). Solving, we find \(B = \begin{bmatrix} 1 & 0 \ -1 & 1 \ 1 & 1 \end{bmatrix}\), verifying that \(AB = I_2\).
03
Understanding Matrix M
Matrix \(M\) is a 3x2 matrix. To find a left-inverse, \(M\) must be full column rank (equal to 2). The columns being linearly independent indicate that \(M\) has full column rank, so a left-inverse exists. However, \(M\) cannot have a right-inverse because it has more rows than columns (i.e., not full row-rank).
04
Finding a Left-Inverse of M
To find the left-inverse of \(M\), consider a 2x3 matrix \(C\) such that \(CM=I_2\). Solve \(\begin{bmatrix} x_{11} & x_{12} & x_{13} \ x_{21} & x_{22} & x_{23} \end{bmatrix} \begin{bmatrix} 1 & 0 \ 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). The solution yields \(C = \begin{bmatrix} 1 & 0 & 0 \ -1 & 1 & 1 \end{bmatrix}\), verifying \(CM=I_2\).
05
Understanding and Analyzing Matrix T
Matrix \(T\) is a 2x2 matrix such that finding a left- or right-inverse requires that \(T\) be invertible, which means its determinant must be non-zero. The determinant \(\det(T) = a^2\), so \(T\) has an inverse as long as \(a eq 0\). Thus, both left-inverse and right-inverse exist if \(a\) is non-zero, and they are identical.
06
Finding the Inverse of T
If \(a eq 0\), the inverse of \(T\) is found by swapping elements across the main diagonal and changing the signs of off-diagonal elements, divided by the determinant. Thus, the inverse of \(T = \begin{bmatrix} a & b \ 0 & a \end{bmatrix}\) is \(\begin{bmatrix} \frac{1}{a} & -\frac{b}{a^2} \ 0 & \frac{1}{a} \end{bmatrix}\). This inverse is both a left-inverse and a right-inverse.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Left-inverse
In linear algebra, a left-inverse of a matrix is a matrix that, when multiplied with the original matrix on the left, yields the identity matrix. Imagine we have a matrix, say a 3x2 matrix like matrix M from our problem. To find its left-inverse, the original matrix must have full column rank. This means the columns should be linearly independent.
For our matrix M:
For our matrix M:
- The columns are linearly independent, indicating a full column rank.
- Thus, a left-inverse exists.
- We found a matrix C such that when C is multiplied by M, it gives the identity matrix I.
Right-inverse
A right-inverse of a matrix is a matrix that, when multiplied with the original matrix on the right, results in the identity matrix. For our matrix A, which is a 2x3 matrix, a right-inverse exists if the matrix has full row rank.
Here鈥檚 what that means for matrix A:
Here鈥檚 what that means for matrix A:
- Matrix A is linearly independent in its rows, achieving full row rank.
- Therefore, it possesses a right-inverse.
- We resolved the matrix equation to find a matrix B, showing that multiplying A by B yields the identity matrix.
Determinant
The determinant is a key property of square matrices (those with the same number of rows and columns). The determinant can tell us if a matrix is invertible 鈥� which means it has an inverse 鈥� if the determinant is non-zero. For the matrix T in our exercise, which is 2x2:
- The determinant of T is calculated as \( ext{det}(T) = a^2 \).
- If \(a eq 0\), the determinant is non-zero, meaning T is invertible.
- Consequently, both a left-inverse and right-inverse exist and are identical.
Linearly Independent
Linear independence is a concept that helps us understand relationships between vectors, and by extension, the rows or columns of a matrix. When we say that vectors (or matrix rows/columns) are linearly independent, it means that no vector can be written as a combination of others.
In our exercise, linearly independent rows or columns determine if:
In our exercise, linearly independent rows or columns determine if:
- A right-inverse exists (like for matrix A with independent rows).
- A left-inverse is possible (like for matrix M with independent columns).
- Simplifies understanding of matrix rank and if/how an inverse can exist.
