Math 270 Final Review II: Is Àº Àº Àº Àº À» À¹ Àª Àª Àª Àª

Math 270 Final Review II 1. Is ຠຠຠຠ໠๠ઠઠઠઠૠé + + a a a a a invertible? Explain.

Determine whether the matrix A, composed of the vectors àº, àº, àº, àº, à» and others, is invertible. To do this, analyze its determinant, rank, and whether its columns are linearly independent. If the determinant is non-zero, the matrix is invertible; if zero, it is singular. Alternatively, check if the columns form a linearly independent set by seeing if the nullity is zero. These are standard approaches rooted in linear algebra fundamentals, such as Rouché–Capelli theorem and properties of determinants.

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The question prompts an investigation into whether a specific matrix A, constructed from certain vectors (not explicitly detailed here), is invertible. In linear algebra, a matrix's invertibility hinges on several equivalent conditions: its determinant being non-zero, its columns (or rows) being linearly independent, and its rank being equal to the size of the matrix. To determine whether matrix A is invertible, the first step is to compute the determinant. If det(A) ≠ 0, this confirms invertibility, indicating that the columns of A span the entire space and are linearly independent. Conversely, det(A) = 0 implies the presence of linear dependence among columns, rendering the matrix non-invertible.

Another essential criterion involves examining the rank of A. If rank(A) equals the number of columns (or size n for an n×n matrix), then A is invertible. The relationship between rank, nullity, and the dimension of the null space (or kernel) also provides insights: an invertible matrix has a nullity of zero, indicating only the trivial solution exists for Ax=0. These properties are consolidated in the Rank-Nullity Theorem, which states that for an m×n matrix A, rank(A) + nullity(A) = n.

Given the specific vectors' elements, one would perform row reduction to echelon form or compute the determinant directly to check for invertibility. If the vectors are explicitly provided, their linear independence can be tested by forming the matrix A with these vectors as columns and assessing the determinants or through Gaussian elimination. Ultimately, if any of these tests—non-zero determinant, full rank, trivial null space—confirm the matrix's properties, then matrix A is invertible; else, it is singular and non-invertible.

References

  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson Education.
  • Strang, G. (2009). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Anton, H., & Rorres, C. (2013). Elementary Linear Algebra. John Wiley & Sons.
  • Leon, S. J., & Roy, H. (2015). Linear Algebra with Applications. Pearson Education.
  • Lay, D. C., Lay, S. R., & McDonald, J. J. (2016). Linear Algebra and Its Applications. Pearson.
  • Halmos, P. R. (1987). A Hilbert Space Problem Book. Springer.
  • Strang, G. (2014). Linear Algebra and Its Applications. Brooks Cole.
  • Axler, S. (2015). Linear Algebra Done Right. Springer.
  • Gallian, J. A. (2017). Contemporary Abstract Algebra. Cengage Learning.
  • Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice-Hall.

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