Invertible Matrix Theorem: Applications

This assignment focuses on the Invertible Matrix Theorem, exploring four of its crucial statements. It demonstrates how these statements apply to a given 3x3 matrix A through row echelon form calculations, linear independence analysis, eigenvalue determination, and determinant evaluation. The aim is to illustrate the interconnectedness and validity of these concepts within the theorem.

Running head: THE INVERTIBLE MATRIX THEOREM 1
The Invertible Matrix Theorem
Name
Institution

THE INVERTIBLE MATRIX THEOREM 2
The Invertible Matrix Theorem
Let then A be a 3 ×3 ¿ where n=3. A=
[ 1 −3 0
−4 11 1
0 7 3 ]
We consider four statements from the invertible Matrix Theorem as shown.
The statements extracted from the Invertible Matrix Theorem include: “Matrix A has n pivot
positions, the columns of the matrix form a linearly independent set, zero is not an eigenvalue of
matrix A, and the determinant of A, det ( A)≠0 .”
Part a
“Matrix A has n pivot positions”
The row echelon form of the Matrix A can be computed as:
R1: “add 4 times the 1st row to the 2nd row” to obtain [1 −3 0
0 −1 1
2 7 3 ]
R2: “add 2 times the 1st row to the 3rd row” to obtain [1 −3 0
0 −1 1
0 13 3 ]
R3: “multiply the 2nd row by -1” to obtain [1 −3 0
0 1 1
0 13 3 ]
R4: add -13 times the 2nd row to 3rd row to obtain [1 −3 0
0 1 −1
0 0 16 ]

THE INVERTIBLE MATRIX THEOREM 3
R5: multiply the third row by 1/16 to obtain [1 −3 0
0 1 −1
0 0 1 ]
Therefore, 3(=n) pivot positions and the statement uphold.
Part b
“The columns of the matrix form a linearly independent set.”
Here, we write the matrix A into a homogenous system of equations
1 c1−4 c2+2 c3=0
−3 c1 +11 c2 +7 c3=0
0 c1 +1 c2 +3 c3=0
Then, transforming the coefficient matrix into the “reduced row echelon form” to obtain
[1 0 0
0 1 0
0 0 1 ], which corresponds to 1 c1=1 c2 =1c3=0. Evidently, each column contains a leading
entry and the matrix has the trivial solution (0,0,0). Therefore, the set is linearly independent and
the statement upholds.
Part c
“Zero is not an eigenvalue of matrix A”
|A|= [ 1 −3 0
−4 11 1
0 7 3 ]

THE INVERTIBLE MATRIX THEOREM 4
The singular matrix, A−λI = [1−λ −3 0
−4 11−λ 1
0 7 3−λ ]=−λ3 +15 λ2−28 λ−16=0
The corresponding eigen values are ( 12.695
2.7613
−0.4564 ) which are non-zero.
Part d
“The determinant of A, det ( A)≠0”
det |A |=
[ 1 −3 0
−4 11 1
0 7 3 ]=1 ( 33−7 ) + 3 ( −12−2 ) +0=−16(≠ 0)
Hence, the four statements (part a through d) obtained from the Invertible Matrix Theorem are all
true.

THE INVERTIBLE MATRIX THEOREM 5
References
Rodman, L. X., & Spitkovsky, I. M. (2005). Analytic roots of invertible matrix
functions. Electronic Journal of Linear Algebra, 13(1). doi:10.13001/1081-3810.1161
Wawro, M. (2015). Reasoning About Solutions in Linear Algebra: the Case of Abraham and the
Invertible Matrix Theorem. International Journal of Research in Undergraduate
Mathematics Education, 1(3), 315-338. doi:10.1007/s40753-015-0017-7