Solving Systems Using Matrix Inverses - Algebra 2

1. What is the identity matrix $I_3$?

   $I_3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$. The 3x3 identity matrix has 1s on the main diagonal and 0s elsewhere, acting as the multiplicative identity.

2. What is $A^{-1}$ for $A=\begin{pmatrix}2&0\\0&5\end{pmatrix}$?

   $A^{-1}=\begin{pmatrix}\frac{1}{2}&0\\0&\frac{1}{5}\end{pmatrix}$. The inverse of a diagonal matrix reciprocates each diagonal entry.

3. What is the relationship between $\det(A)$ and the number of solutions to $A\vec{x}=\vec{b}$?

   $\det(A)\neq 0$ implies a unique solution for every $\vec{b}$. A non-zero determinant confirms invertibility, guaranteeing a unique solution for every $\vec{b}$.

4. Identify whether $A=\begin{pmatrix}1&2\\2&4\end{pmatrix}$ is invertible.

   Not invertible because $\det(A)=0$. A zero determinant confirms the matrix is singular.

5. What does it mean for the columns of $A$ if $A$ is invertible?

   The columns are linearly independent. Invertibility requires the columns to form a linearly independent set.

6. What does it indicate if row reduction of $[A\mid I]$ cannot make the left side $I$?

   $A$ is not invertible, so $A^{-1}$ does not exist. Inability to reach the identity on the left indicates the matrix is singular.

7. What is the augmented matrix form of the system $A\vec{x}=\vec{b}$?

   $[A\mid \vec{b}]$. The augmented matrix combines coefficients and constants for row reduction.

8. What is the identity matrix $I_2$?

   $I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$. The 2x2 identity matrix has 1s on the main diagonal and 0s elsewhere, serving as the multiplicative identity.

9. What matrix equation represents a linear system in matrix form?

   $A\vec{x}=\vec{b}$. This form represents the linear system as a matrix-vector product equaling the constant vector.

10. What is the inverse-matrix solution to $A\vec{x}=\vec{b}$ when $A$ is invertible?

    $\vec{x}=A^{-1}\vec{b}$. Multiplying both sides on the left by $A^{-1}$ isolates $\vec{x}$ as the unique solution.

11. Identify the correct criterion: $A$ is invertible iff $\det(A)=0$ or $\det(A)\neq 0$?

    $A$ is invertible iff $\det(A)\neq 0$. Invertibility requires a non-zero determinant.

12. What is the inverse of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$ when all $d_i\neq 0$?

    $D^{-1}=\mathrm{diag}(\frac{1}{d_1},\dots,\frac{1}{d_n})$. Each diagonal entry is reciprocated to ensure the product is the identity matrix.

13. Find $\det\!\left(\begin{pmatrix}1&0&0\\2&3&0\\4&5&6\end{pmatrix}\right)$.

    $18$. For upper triangular matrices, the determinant is the product of diagonal entries.

14. Identify whether $A=\begin{pmatrix}4&-2\\6&-3\end{pmatrix}$ is invertible.

    Not invertible because $\det(A)=0$. A zero determinant indicates linear dependence and non-invertibility.

15. What does it mean if $A\vec{x}=\vec{b}$ has a unique solution for every $\vec{b}$?

    $A$ is invertible. Invertibility ensures the system has a unique solution for any right-hand side vector.

16. What is the inverse of a scalar multiple $kA$ when $k\neq 0$ and $A$ is invertible?

    $(kA)^{-1}=\frac{1}{k}A^{-1}$. The inverse scales by the reciprocal of $k$ to counteract the scalar multiple.

17. What is the determinant of $A=\begin{pmatrix}a&0\\0&d\end{pmatrix}$?

    $\det(A)=ad$. For diagonal matrices, the determinant is the product of diagonal entries.

18. Identify the correct solution form for $A\vec{x}=\vec{b}$ when $A$ is invertible: $\vec{x}=A\vec{b}$ or $\vec{x}=A^{-1}\vec{b}$?

    $\vec{x}=A^{-1}\vec{b}$. The correct form uses the inverse matrix multiplied on the left by $\vec{b}$.

19. Compute $(3A)^{-1}$ if $A^{-1}=M$ and $3\neq 0$.

    $(3A)^{-1}=\frac{1}{3}M$. The scalar 3 requires dividing the inverse by 3.

20. What is the determinant test for invertibility of a square matrix $A$?

    $A$ is invertible iff $\det(A)\neq 0$. A non-zero determinant indicates the matrix has full rank and is thus invertible.

21. What is $A^{-1}$ for $A=\begin{pmatrix}1&0&0\\0&-2&0\\0&0&4\end{pmatrix}$?

    $A^{-1}=\begin{pmatrix}1&0&0\\0&-\frac{1}{2}&0\\0&0&\frac{1}{4}\end{pmatrix}$. Reciprocate each diagonal entry to form the inverse.

22. Compute $\det\!\left(\begin{pmatrix}5&0&0\\0&1&0\\0&0&-2\end{pmatrix}\right)$.

    $-10$. The determinant of a diagonal matrix is the product of its diagonal entries.

23. Solve $A\vec{x}=\vec{b}$ if $A^{-1}=\begin{pmatrix}1&0\\-2&1\end{pmatrix}$ and $\vec{b}=\begin{pmatrix}3\\5\end{pmatrix}$.

    $\vec{x}=\begin{pmatrix}3\\-1\end{pmatrix}$. Matrix multiplication of $A^{-1}$ by $\vec{b}$ gives the solution vector.

24. What is $\det\!\left(\begin{pmatrix}3&1\\2&4\end{pmatrix}\right)$?

    $10$. Computed using the 2x2 determinant formula $ad-bc$.

25. What is the inverse of the inverse of an invertible matrix $A$?

    $(A^{-1})^{-1}=A$. Inverting the inverse restores the original matrix, as they are multiplicative inverses.

26. What does it mean for the system $A\vec{x}=\vec{b}$ if $A$ is not invertible?

    The system has either no solution or infinitely many solutions. Non-invertible matrices lead to systems that are either inconsistent or have infinite solutions.

27. Solve using an inverse: $\begin{pmatrix}1&0\\0&2\end{pmatrix}\vec{x}=\begin{pmatrix}5\\6\end{pmatrix}$.

    $\vec{x}=\begin{pmatrix}5\\3\end{pmatrix}$. Multiplying the inverse matrix by the vector yields the solution.

28. Identify the correct statement: $A^{-1}\vec{b}$ or $\vec{b}A^{-1}$ solves $A\vec{x}=\vec{b}$?

    $\vec{x}=A^{-1}\vec{b}$. $A^{-1}\vec{b}$ is the correct order for matrix-vector multiplication to solve the system.

29. What is the inverse of the product $AB$ when $A$ and $B$ are invertible?

    $(AB)^{-1}=B^{-1}A^{-1}$. The inverse of a product requires reversing the order of the individual inverses.

30. What is the determinant of a triangular matrix in terms of its diagonal entries?

    $\det(A)$ equals the product of the diagonal entries. For triangular matrices, the determinant is the product of the diagonal elements.

31. What is the determinant of $A=\begin{pmatrix}a&b\\0&d\end{pmatrix}$?

    $\det(A)=ad$. For upper triangular matrices, the determinant equals the product of diagonals.

32. Which matrix must have the same size as $A$ to form $[A\mid I]$ for an $n\times n$ matrix?

    The identity matrix $I_n$. $I_n$ is square $n\times n$ to match $A$'s dimensions for augmentation.

33. What is $\det(A^{-1})$ in terms of $\det(A)$ for invertible $A$?

    $\det(A^{-1})=\frac{1}{\det(A)}$. This follows from the multiplicative property of determinants and $\det(I)=1$.

34. What is the determinant of the identity matrix $I_n$?

    $\det(I_n)=1$. It is the product of the diagonal entries, all of which are 1.

35. Identify whether $A=\begin{pmatrix}1&0&0\\0&0&2\\0&3&0\end{pmatrix}$ is invertible using $\det(A)$.

    Invertible because $\det(A)=-6\neq 0$. The non-zero determinant confirms the matrix is invertible.

36. Compute $(AB)^{-1}$ if $A^{-1}=P$ and $B^{-1}=Q$ for invertible $A,B$.

    $(AB)^{-1}=QP$. Given $A^{-1}=P$ and $B^{-1}=Q$, the product inverse is $B^{-1}A^{-1}$.

37. What is $\det(AB)$ in terms of $\det(A)$ and $\det(B)$ for square matrices?

    $\det(AB)=\det(A)\det(B)$. The determinant function is multiplicative for square matrices.

38. Find $\left(\begin{pmatrix}0&2\\-3&1\end{pmatrix}\right)^{-1}$.

    $\begin{pmatrix}\frac{1}{6}&-\frac{1}{3}\\frac{1}{2}&0\end{pmatrix}$. The inverse is scaled by $1/\det(A)$ using the adjugate matrix.

39. What is the solution of $A\vec{x}=\vec{0}$ when $A$ is invertible?

    $\vec{x}=\vec{0}$. Invertible $A$ implies only the trivial solution for the homogeneous system.

40. What is $A^{-1}$ if $A$ is the identity matrix $I_n$?

    $I_n^{-1}=I_n$. The identity matrix multiplied by itself yields itself, satisfying the inverse definition.

41. What technology-based method is commonly used to find $A^{-1}$ for $3\times 3$ or larger matrices?

    Use a calculator or software to compute $A^{-1}$ or solve $A\vec{x}=\vec{b}$. Technology is recommended for efficient computation of inverses or solutions for larger matrices.

42. Solve using an inverse: $\begin{pmatrix}2&0\\0&-4\end{pmatrix}\vec{x}=\begin{pmatrix}8\\12\end{pmatrix}$.

    $\vec{x}=\begin{pmatrix}4\\-3\end{pmatrix}$. The inverse scales each equation to solve for the variables.

43. What does it mean if a row of zeros appears on the left side when row-reducing $[A\mid I]$?

    $A$ is singular, so $A^{-1}$ does not exist. A zero row indicates linear dependence, meaning the matrix is singular.

44. What does $A\vec{x}=\vec{b}$ represent when $A$ is $3\times 3$?

    A system of $3$ linear equations in $3$ unknowns. The dimensions of $A$ determine the number of equations and unknowns.

45. What does it mean for a square matrix $A$ to be invertible?

    $A^{-1}$ exists such that $AA^{-1}=I$ and $A^{-1}A=I$. A square matrix is invertible if there exists a unique matrix that, when multiplied on either side, yields the identity matrix.

46. What condition on $a,b,c,d$ makes $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ non-invertible?

    $ad-bc=0$. This condition makes the determinant zero, rendering the matrix singular and non-invertible.

47. What is the determinant of a diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$?

    $\det(D)=d_1d_2\cdots d_n$. The determinant is the product of its diagonal entries.

48. What matrix is used to compute $A^{-1}$ by row reduction for an $n\times n$ matrix $A$?

    Augment $[A\mid I_n]$ and row-reduce to $[I_n\mid A^{-1}]$. Row operations transform $A$ to $I_n$, applying the same to $I_n$ to obtain $A^{-1}$.

49. Identify the matrix equation you should solve to find $A^{-1}$ using row reduction.

    Row-reduce $[A\mid I_n]$ to $[I_n\mid A^{-1}]$. Row reduction of the augmented matrix yields the inverse on the right.

50. What is the inverse of a $2\times 2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ when it exists?

    $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. The formula scales the adjugate matrix by the reciprocal of the determinant to satisfy $AA^{-1}=I$.

51. What is the transpose-inverse rule for an invertible matrix $A$?

    $(A^T)^{-1}=(A^{-1})^T$. The transpose of the inverse equals the inverse of the transpose for invertible matrices.

52. What equation defines the inverse matrix $A^{-1}$ of a square matrix $A$?

    $AA^{-1}=I$ (equivalently $A^{-1}A=I$). The inverse satisfies the multiplicative identity property when multiplied on the left or right by the original matrix.

53. Compute $\det\!\left(\begin{pmatrix}0&2\\-3&1\end{pmatrix}\right)$.

    $6$. Apply the 2x2 determinant formula $ad-bc$.

54. Find $\left(\begin{pmatrix}1&2\\3&4\end{pmatrix}\right)^{-1}$.

    $\begin{pmatrix}-2&1\\frac{3}{2}&-\frac{1}{2}\end{pmatrix}$. The 2x2 inverse formula scales the adjugate by $1/\det(A)$.

55. What is the matrix size of $A$ in $A\vec{x}=\vec{b}$ for a system of $n$ equations in $n$ unknowns?

    $A$ is $n\times n$. A square $n\times n$ matrix corresponds to $n$ equations in $n$ variables.

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