Zero and Identity Matrices and Determinants - Geometry

1. If $\det(A)\ne 0$, what must be true about the equation $AX=I$?

   It has the unique solution $X=A^{-1}$. Nonzero determinant guarantees unique solution exists.

2. If $\det(A)=\frac{1}{4}$, what is $\det(A^{-1})$?

   $4$. Inverse determinant is reciprocal: $\frac{1}{1/4} = 4$.

3. What is $A+I_n$ called when $A$ is $n\times n$?

   It is the matrix sum of $A$ and $I_n$ (defined entrywise). Matrix addition performed entrywise on compatible matrices.

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

   $A$ is invertible if and only if $\det(A)\ne 0$. Nonzero determinant is the invertibility criterion.

5. What is $I_3A$ if $A$ is a $3\times 3$ matrix?

   $I_3A=A$. Identity matrix leaves $A$ unchanged in multiplication.

6. If $A$ is invertible, what is $\det(A)\det(A^{-1})$?

   $\det(A)\det(A^{-1})=1$. From $\det(AA^{-1}) = \det(I) = 1$.

7. Identify the correct statement about $I_n$ and $0_n$ under multiplication.

   $AI_n=A$ and $A0_n=0_n$ (for $n\times n$ matrix $A$). Identity acts as 1, zero acts as 0.

8. If $A$ is invertible, what is $\det(A^{-1})$ in terms of $\det(A)$?

   $\det(A^{-1})=\frac{1}{\det(A)}$. Inverse determinant is reciprocal of original.

9. What is $\det\left(\begin{pmatrix}0&0\\0&0\end{pmatrix}\right)$?

   $0$. Determinant of $2 \times 2$ zero matrix.

10. What is the determinant of a $2\times 2$ matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

    $\det=ad-bc$. Standard $2 \times 2$ determinant formula.

11. Which matrix is the additive identity for $2\times 2$ matrices?

    $\begin{pmatrix}0&0\\0&0\end{pmatrix}$. Standard form of $2 \times 2$ zero matrix.

12. What does $\det(A)=0$ imply about the matrix $A$?

    $A$ is not invertible (it is singular). Zero determinant means no multiplicative inverse exists.

13. If $\det(A)=3$ and $\det(B)=5$ for $2\times 2$ matrices, what is $\det(AB)$?

    $15$. Using multiplicative property: $3 \times 5 = 15$.

14. Determine whether $A=\begin{pmatrix}1&2\\3&5\end{pmatrix}$ is invertible using $\det(A)$.

    Invertible because $\det(A)=-1\ne 0$. $\det = 1(5) - 2(3) = -1 \ne 0$.

15. Find $\det\left(\begin{pmatrix}3&1\\2&4\end{pmatrix}\right)$.

    $10$. Using $\det = ad - bc = 3(4) - 1(2) = 10$.

16. What is $\det(0_n)$ for any positive integer $n$?

    $\det(0_n)=0$. Zero matrix has zero determinant always.

17. What is $0_{m\times n}\cdot B$ when $B$ is $n\times p$?

    $0_{m\times n}B=0_{m\times p}$. Zero matrix multiplication yields zero matrix.

18. Which determinant value guarantees that a $2\times 2$ matrix has no inverse?

    $\det(A)=0$. Zero determinant prevents inverse from existing.

19. What matrix equation defines the identity matrix as a neutral element for multiplication?

    For any $A$, $AI=IA=A$ (sizes compatible). Identity matrix leaves other matrices unchanged.

20. What is the inverse formula for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ when $\det(A)\ne 0$?

    $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. Derived from determinant formula and cofactor method.

21. What is the result of multiplying any matrix $A$ by a compatible zero matrix?

    $A^0=0$ and $0A=0$ (with compatible sizes). Zero matrix annihilates in multiplication.

22. If $\det(A)=-2$ for a $3\times 3$ matrix, what is $\det(2A)$?

    $2^3\det(A)=-16$. Scalar multiplication: $2^3 \times (-2) = -16$.

23. What is $\det(kA)$ for an $n\times n$ matrix $A$ and scalar $k$?

    $\det(kA)=k^n\det(A)$. Scalar multiplication raises to matrix dimension power.

24. What is the relationship between $\det(A)$ and $\det(A^T)$ for a square matrix $A$?

    $\det(A^T)=\det(A)$. Transpose preserves determinant value.

25. What is the additive identity property for matrices?

    For any $A$, $A+0=A$. Zero matrix acts like zero in addition.

26. What is $A+0_{m\times n}$ for an $m\times n$ matrix $A$?

    $A+0_{m\times n}=A$. Adding zero matrix leaves $A$ unchanged.

27. What is $\det\left(\begin{pmatrix}7&0\\0&2\end{pmatrix}\right)$?

    $14$. Diagonal matrix determinant is product: $7 \times 2$.

28. What is $\det\left(\begin{pmatrix}1&0\\0&1\end{pmatrix}\right)$?

    $1$. Determinant of $2 \times 2$ identity matrix.

29. What is $A- A$ for any matrix $A$?

    $A-A=0$. Subtracting a matrix from itself gives zero.

30. What is the additive inverse property for matrices?

    For any $A$, $A+(-A)=0$. Adding a matrix to its negative gives zero.

31. What is $\det\left(\begin{pmatrix}5&1\\0&3\end{pmatrix}\right)$?

    $15$. Upper triangular: product of diagonal entries.

32. What is $A\cdot 0_{n\times p}$ when $A$ is $m\times n$?

    $A0_{n\times p}=0_{m\times p}$. Multiplying by zero matrix gives zero result.

33. If $A$ is invertible, what must be true about $\det(A)$?

    $\det(A)\ne 0$. Invertible matrices have nonzero determinants.

34. What does $\det(A)\ne 0$ imply about the matrix $A$?

    $A$ has a multiplicative inverse $A^{-1}$. Nonzero determinant guarantees invertibility.

35. What is the determinant of a matrix with two identical rows?

    $\det(A)=0$. Identical rows make matrix linearly dependent.

36. What happens to $\det(A)$ if a multiple of one row is added to another row?

    The determinant is unchanged. Row addition preserves determinant value.

37. What is the determinant of a triangular $n\times n$ matrix?

    $\det(A)$ equals the product of the diagonal entries. Triangular matrices have diagonal-only nonzero contribution.

38. What happens to $\det(A)$ if two rows of $A$ are swapped?

    The determinant changes sign: $\det(\text{swapped})=-\det(A)$. Row swap operation flips determinant sign.

39. What is the determinant of a matrix with a row of all zeros?

    $\det(A)=0$. Zero row makes matrix singular (not invertible).

40. What is $AI_4$ if $A$ is a $4\times 4$ matrix?

    $AI_4=A$. Right multiplication by identity preserves matrix.

41. What is the determinant of a product of square matrices $A$ and $B$ of the same size?

    $\det(AB)=\det(A)\det(B)$. Determinant is multiplicative for square matrices.

42. What is the $n\times n$ zero matrix, written $0_n$?

    $0_n$ is the $n\times n$ matrix with every entry $0$. Each entry is zero, forming the additive identity.

43. Identify the additive identity in the set of all $m\times n$ matrices.

    The $m\times n$ zero matrix $0$. Zero matrix is unique additive neutral element.

44. What is the multiplicative identity property for square matrices?

    For any $n\times n$ matrix $A$, $AI_n=I_nA=A$. Identity matrix acts like 1 in multiplication.

45. What is the determinant of a diagonal $n\times n$ matrix $\mathrm{diag}(d_1,\dots,d_n)$?

    $\det=d_1d_2\cdots d_n$. Only diagonal entries contribute to determinant.

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

    There exists $A^{-1}$ such that $AA^{-1}=A^{-1}A=I$. The inverse undoes multiplication to give identity.

47. Determine whether $A=\begin{pmatrix}2&1\\4&2\end{pmatrix}$ is invertible using $\det(A)$.

    Not invertible because $\det(A)=0$. $\det = 2(2) - 1(4) = 0$, so not invertible.

48. What is the $n\times n$ identity matrix, written $I_n$?

    $I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal of ones makes it the multiplicative identity.

49. What happens to $\det(A)$ if one row is multiplied by $k$?

    The determinant is multiplied by $k$. Scalar row multiplication scales determinant.

50. Which matrix is the multiplicative identity for $2\times 2$ matrices?

    $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Standard form of $2 \times 2$ identity matrix.

51. What is $\det(I_n)$ for any positive integer $n$?

    $\det(I_n)=1$. Identity matrix has determinant 1 by definition.

52. If $\det(A)=0$, what is the correct conclusion about $A^{-1}$?

    $A^{-1}$ does not exist. Zero determinant means matrix is not invertible.

53. Identify the multiplicative identity in the set of all $n\times n$ matrices.

    The $n\times n$ identity matrix $I_n$. Identity matrix is unique multiplicative neutral element.

54. Compute $\det\left(\begin{pmatrix}2&0&0\\0&-1&0\\0&0&4\end{pmatrix}\right)$.

    $-8$. Diagonal matrix: product $2 \times (-1) \times 4$.

Study these flashcards