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Section4.4Matrix Algebra

SubsectionMatrix Multiplication

SubsectionMatrix Multiplication Is Function Composition

SubsectionThe Inverse of a Matrix

SubsectionExercises

1

Compute the following sum. \begin{equation*} \left[ \begin{array}{rrr} 3 \amp -2 \amp 5 \\ 11 \amp 0 \amp -1 \\ 6 \amp 4 \amp 6 \\ \end{array} \right] + \left[ \begin{array}{rrr} -4 \amp 3 \amp 1 \\ 3 \amp -2 \amp 2 \\ 5 \amp 15 \amp -1 \\ \end{array} \right] \end{equation*}

2

Simplify the following linear combination of two, \(2\times 2\) matrices. \begin{equation*} 2 \left[ \begin{array}{rrr} 5 \amp 3 \\ 1 \amp -1 \end{array} \right] + (-3) \left[ \begin{array}{rrr} -4 \amp 1 \\ 1 \amp -2 \end{array} \right] \end{equation*}

3

If the multiplication is defined, compute the product, otherwise write “UNDEFINED” on your paper.

  1. \begin{equation*} \left[ \begin{array}{rrr} -2 \amp 6 \\ 0 \amp 2 \end{array} \right] \left[ \begin{array}{rrr} 1 \amp 3 \amp -3 \\ -1 \amp 4 \amp 0 \end{array} \right] \end{equation*}

  2. \begin{equation*} \left[ \begin{array}{rrr} 3 \amp 5 \amp -4 \end{array} \right] \left[ \begin{array}{rrr} 2 \\ 1 \\ 4 \end{array} \right] \end{equation*}

  3. \begin{equation*} \left[ \begin{array}{rrr} 2 \\ 1 \\ 4 \end{array} \right] \left[ \begin{array}{rrr} 3 \amp 5 \amp -4 \end{array} \right] \end{equation*}

  4. \begin{equation*} \left[ \begin{array}{rr} 2 \amp 3 \\ 1 \amp -3\\ 0 \amp 4 \end{array} \right] \left[ \begin{array}{rrr} 1 \amp 0 \amp -2 \\ 8 \amp 5 \amp 1 \\ 1 \amp -1 \amp 4 \end{array} \right] \end{equation*}

4

Compute \(A^{-1}\text{.}\) And show that \(AA^{-1} = I\) and \(A^{-1}A = I\text{.}\) \begin{equation*} \left[ \begin{array}{rrr} 4 \amp 0 \\ 3 \amp 5 \end{array} \right] \end{equation*}

Answer
5

Compute the inverse by augmenting the matrix on the right-hand side with the identity matrix and performing elementary row operations on the augmented matrix until the left side is the identity. \begin{equation*} A = \left[ \begin{array}{rrr} 2 \amp 3 \amp 0 \\ 1 \amp 1 \amp -1 \\ 0 \amp 2 \amp 2 \end{array} \right] \end{equation*}

Answer
6

Compute the inverse by augmenting the matrix on the right-hand side with the identity matrix and performing elementary row operations on the augmented matrix until the left side is the identity. \begin{equation*} A = \left[ \begin{array}{rrr} 1 \amp 5 \amp 1 \\ 2 \amp 5 \amp 0 \\ 2 \amp 7 \amp 1 \end{array} \right] \end{equation*}

7

Compute the inverse by augmenting the matrix on the right-hand side with the identity matrix and performing elementary row operations on the augmented matrix until the left side is the identity. \begin{equation*} A = \left[ \begin{array}{rrrr} 0 \amp 0 \amp 1 \amp 0\\ 1 \amp 0 \amp 0 \amp 0\\ 0 \amp 1 \amp 2 \amp 0\\ 3 \amp 0 \amp 0 \amp 1 \end{array} \right] \end{equation*}

8

Compute the inverse by augmenting the matrix on the right-hand side with the identity matrix and performing elementary row operations on the augmented matrix until the left side is the identity. \begin{equation*} \left[ \begin{array}{rrr} 1 \amp 4 \amp 3 \\ 1 \amp 4 \amp 5 \\ 2 \amp 5 \amp 1 \end{array} \right] \end{equation*}

Answer
9

Suppose that \(A, B, C\) are invertible \(n\times n\) matrices. Use the definition of the inverse of a matrix to show that: \begin{equation*} (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \end{equation*}

10

A diagonal matrix is a matrix with nonzero entries down its diagonal and zeros everywhere else. Assuming \(a_1, a_2, \ldots, a_n \ne 0\text{,}\) what is the inverse of: \begin{equation*} \left[ \begin{array}{rrrrr} a_1 \amp 0 \amp 0 \amp \cdots \amp 0\\ 0 \amp a_2\amp 0 \amp \cdots \amp 0 \\ 0 \amp 0 \amp a_3 \amp \amp \vdots \\ \vdots \amp \vdots \amp \amp \ddots \amp 0 \\ 0 \amp 0 \amp \cdots \amp 0 \amp a_n \end{array} \right] \end{equation*} If you don't know the answer, start small by computing the inverse of a \(2\times 2\) diagonal matrix, and then try to discover the pattern.