Finding Elements Of Matrix D Given CD = I

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In mathematics, particularly in linear algebra, understanding matrix operations is crucial. One common operation is finding the inverse of a matrix. This article delves into a specific problem: given that CD = I, where C is a known matrix and I is the identity matrix, we aim to find the elements of matrix D. This exploration will provide a comprehensive understanding of how to compute the inverse matrix and the underlying principles of matrix algebra. Let's break down the problem step by step.

Problem Statement

We are given that:

CD=ICD = I

Where:

C=[−241103−23−1]C = \begin{bmatrix} -2 & 4 & 1 \\ 1 & 0 & 3 \\ -2 & 3 & -1 \end{bmatrix}

D=[abcdefghi]D = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}

I=[100010001]I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

The task is to find the elements a, b, c, d, e, f, g, h, i of matrix D. Since CD = I, matrix D is the inverse of matrix C, denoted as C⁻¹.

Understanding Matrix Inversion

What is an Inverse Matrix?

In linear algebra, the inverse of a square matrix A, denoted as A⁻¹, is a matrix that, when multiplied by A, yields the identity matrix I. That is:

A∗A−1=A−1∗A=IA * A^{-1} = A^{-1} * A = I

Not all matrices have an inverse. A matrix is invertible (or non-singular) if its determinant is non-zero. If the determinant is zero, the matrix is singular and does not have an inverse.

Methods to Find the Inverse of a Matrix

There are several methods to find the inverse of a matrix, including:

  1. Gaussian Elimination (Row Reduction): This method involves augmenting the matrix with the identity matrix and performing row operations until the original matrix becomes the identity matrix. The augmented part then becomes the inverse.

  2. Using the Adjugate (Adjoint) Matrix: The inverse of a matrix A can be found using the formula:

    A−1=1det(A)∗adj(A)A^{-1} = \frac{1}{det(A)} * adj(A)

    Where det(A) is the determinant of A, and adj(A) is the adjugate (or adjoint) of A.

  3. Using Software Tools: Various software tools and programming libraries (e.g., NumPy in Python, MATLAB) can compute the inverse of a matrix.

In this article, we will use the method of Gaussian elimination (row reduction) to find the inverse of matrix C.

Step-by-Step Solution Using Gaussian Elimination

Step 1: Augment the Matrix

We start by augmenting matrix C with the identity matrix I:

[C∣I]=[−241∣100103∣010−23−1∣001][C | I] = \begin{bmatrix} -2 & 4 & 1 &|& 1 & 0 & 0 \\ 1 & 0 & 3 &|& 0 & 1 & 0 \\ -2 & 3 & -1 &|& 0 & 0 & 1 \end{bmatrix}

Step 2: Perform Row Operations

Our goal is to transform the left side of the augmented matrix into the identity matrix using elementary row operations. We perform the following operations:

  1. Swap Row 1 and Row 2:

    R1↔R2R_1 \leftrightarrow R_2

    [103∣010−241∣100−23−1∣001]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ -2 & 4 & 1 &|& 1 & 0 & 0 \\ -2 & 3 & -1 &|& 0 & 0 & 1 \end{bmatrix}

  2. Add 2 times Row 1 to Row 2:

    R2→R2+2R1R_2 \rightarrow R_2 + 2R_1

    [103∣010047∣120−23−1∣001]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 4 & 7 &|& 1 & 2 & 0 \\ -2 & 3 & -1 &|& 0 & 0 & 1 \end{bmatrix}

  3. Add 2 times Row 1 to Row 3:

    R3→R3+2R1R_3 \rightarrow R_3 + 2R_1

    [103∣010047∣120035∣021]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 4 & 7 &|& 1 & 2 & 0 \\ 0 & 3 & 5 &|& 0 & 2 & 1 \end{bmatrix}

  4. Multiply Row 2 by 1/4:

    R2→14R2R_2 \rightarrow \frac{1}{4}R_2

    [103∣0100174∣14120035∣021]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 1 & \frac{7}{4} &|& \frac{1}{4} & \frac{1}{2} & 0 \\ 0 & 3 & 5 &|& 0 & 2 & 1 \end{bmatrix}

  5. Subtract 3 times Row 2 from Row 3:

    R3→R3−3R2R_3 \rightarrow R_3 - 3R_2

    [103∣0100174∣1412000−14∣−34121]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 1 & \frac{7}{4} &|& \frac{1}{4} & \frac{1}{2} & 0 \\ 0 & 0 & -\frac{1}{4} &|& -\frac{3}{4} & \frac{1}{2} & 1 \end{bmatrix}

  6. Multiply Row 3 by -4:

    R3→−4R3R_3 \rightarrow -4R_3

    [103∣0100174∣14120001∣3−2−4]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 1 & \frac{7}{4} &|& \frac{1}{4} & \frac{1}{2} & 0 \\ 0 & 0 & 1 &|& 3 & -2 & -4 \end{bmatrix}

  7. Subtract 7/4 times Row 3 from Row 2:

    R2→R2−74R3R_2 \rightarrow R_2 - \frac{7}{4}R_3

    [103∣010010∣−547001∣3−2−4]\begin{bmatrix} 1 & 0 & 3 &|& 0 & 1 & 0 \\ 0 & 1 & 0 &|& -5 & 4 & 7 \\ 0 & 0 & 1 &|& 3 & -2 & -4 \end{bmatrix}

  8. Subtract 3 times Row 3 from Row 1:

    R1→R1−3R3R_1 \rightarrow R_1 - 3R_3

    [100∣−9712010∣−547001∣3−2−4]\begin{bmatrix} 1 & 0 & 0 &|& -9 & 7 & 12 \\ 0 & 1 & 0 &|& -5 & 4 & 7 \\ 0 & 0 & 1 &|& 3 & -2 & -4 \end{bmatrix}

Step 3: Identify the Inverse Matrix

Now that the left side of the augmented matrix is the identity matrix, the right side is the inverse matrix D:

D=C−1=[−9712−5473−2−4]D = C^{-1} = \begin{bmatrix} -9 & 7 & 12 \\ -5 & 4 & 7 \\ 3 & -2 & -4 \end{bmatrix}

Thus, the elements of matrix D are:

  • a = -9, b = 7, c = 12
  • d = -5, e = 4, f = 7
  • g = 3, h = -2, i = -4

Verification

To verify that D is indeed the inverse of C, we can multiply C and D:

CD=[−241103−23−1]∗[−9712−5473−2−4]CD = \begin{bmatrix} -2 & 4 & 1 \\ 1 & 0 & 3 \\ -2 & 3 & -1 \end{bmatrix} * \begin{bmatrix} -9 & 7 & 12 \\ -5 & 4 & 7 \\ 3 & -2 & -4 \end{bmatrix}

CD=[(−2)(−9)+4(−5)+1(3)(−2)(7)+4(4)+1(−2)(−2)(12)+4(7)+1(−4)1(−9)+0(−5)+3(3)1(7)+0(4)+3(−2)1(12)+0(7)+3(−4)−2(−9)+3(−5)+(−1)(3)−2(7)+3(4)+(−1)(−2)−2(12)+3(7)+(−1)(−4)]CD = \begin{bmatrix} (-2)(-9) + 4(-5) + 1(3) & (-2)(7) + 4(4) + 1(-2) & (-2)(12) + 4(7) + 1(-4) \\ 1(-9) + 0(-5) + 3(3) & 1(7) + 0(4) + 3(-2) & 1(12) + 0(7) + 3(-4) \\ -2(-9) + 3(-5) + (-1)(3) & -2(7) + 3(4) + (-1)(-2) & -2(12) + 3(7) + (-1)(-4) \end{bmatrix}

CD=[18−20+3−14+16−2−24+28−4−9+0+97+0−612+0−1218−15−3−14+12+2−24+21+4]CD = \begin{bmatrix} 18 - 20 + 3 & -14 + 16 - 2 & -24 + 28 - 4 \\ -9 + 0 + 9 & 7 + 0 - 6 & 12 + 0 - 12 \\ 18 - 15 - 3 & -14 + 12 + 2 & -24 + 21 + 4 \end{bmatrix}

CD=[100010001]=ICD = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I

Thus, CD = I, confirming that D is the inverse of C.

Conclusion

In this article, we successfully found the elements of matrix D given that CD = I and matrix C is known. We used Gaussian elimination to determine that matrix D is the inverse of matrix C. This process not only helps in solving mathematical problems but also provides a strong foundation in linear algebra concepts. Understanding matrix inversion is crucial in various fields such as computer graphics, engineering, and data analysis. The ability to manipulate matrices and find their inverses allows for solving systems of linear equations, performing transformations, and much more.

The inverse matrix D is:

D=[−9712−5473−2−4]D = \begin{bmatrix} -9 & 7 & 12 \\ -5 & 4 & 7 \\ 3 & -2 & -4 \end{bmatrix}

This step-by-step guide should help anyone looking to understand and compute matrix inverses, reinforcing the practical applications of linear algebra.