Find the determinants of the linear transformations in Exercises 17through .
a. For an invertiblen×n matrixA and an arbitraryn×n matrix B, show that rref[A∣AB]=[In∣B]rref[A∣AB]=[In∣B].Hint: The left part ofrref[A∣AB] is rref(A)=In. Write rref [A∣AB]=[In∣M]; we have to show thatM=B . To demonstrate this, note that the columns of matrix [B-In]are in the kernel of[A∣AB] and therefore in the kernel of [In∣M].b. What does the formula rref[A∣AB]=[In∣B]tell you if B=A-1
Consider two distinct points[a1a2] and[b1b2] in the plane. Explain why the solutions[x1x2] of the equationdet[111x1a1b1x2a2b2]=0 form a line and why this line goes through the two points[a1a2] and [b1b2]
An8×8matrix fails to be invertible if (and only if) its determinant is nonzero.
Consider a n×n matrixA=[v→1v→1....v→n] . What is the relationship between the product ||v→1||||v→2||...||v→n|| and |detA|When is|detA|=||v→1||||v→2||...||v→n||
Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.
For the matrices A in Exercisethroughfind closed formulas for At where t is an arbitrary positive integer. follow the strategy. outlined in Theoremand illustrated in Example.in Exercisethroughfeel free to use technolog
Find all 2x2matrices for which is an eigenvector with associated eigenvalue -1.
Consider a linear transformation T(x→)=Ax→ from ℝ2toℝ2 . Suppose for two vectors v→1 and v→2in ℝ2 we have T(v→1)=3v→1androle="math" localid="1660719222222" T(v→2)=4v→2 . What can you say about det A ? Justify your answer carefu
Consider a 4×4matrix A with rowsv→1,v2→,v3→,v4→. If det(A)=8, find the determinants in Exercises 11 through 16.
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