Zero and Identity Matrices and Determinants - Precalculus

1. What is $A^0$ if $A$ is any $4 \times 2$ matrix and $0$ is the $2 \times 3$ zero matrix?

   $A^0 = 0$ (the $4 \times 3$ zero matrix). Product dimensions are $4 \times 3$, all entries zero.

2. If a square matrix $A$ has an inverse, what must be true about $\det(A)$?

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

3. What is the value of $0_{m \times n}B$ when $B$ is $n \times p$ and the product is defined?

   $0_{m \times p}$. Zero matrix times any matrix gives zero matrix.

4. What is the value of $AI_n$ when $A$ is an $m \times n$ matrix?

   $A$. Right multiplication by identity preserves the matrix.

5. What is the additive identity matrix for $m \times n$ matrices?

   The zero matrix $0_{m \times n}$. All entries are zero, acts like 0 in real number addition.

6. What is the definition of an inverse matrix $A^{-1}$ for an $n \times n$ matrix $A$?

   $AA^{-1} = I_n$ and $A^{-1}A = I_n$. Inverse matrix multiplied by original yields identity.

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

   $1$. Identity matrix always has determinant 1.

8. What is the $n \times n$ identity matrix $I_n$ in terms of its entries?

   $I_n$ has $1$ on the main diagonal and $0$ elsewhere. Diagonal entries are 1, creating the multiplicative identity.

9. Compute $\det\!\left(\begin{pmatrix}2&1\\4&2\end{pmatrix}\right)$.

   $0$. $2(2)-1(4)=4-4=0$; rows are proportional.

10. What equation defines the multiplicative identity property for square matrices $A$ and $I$?

    $AI = IA = A$. Multiplying by identity leaves square matrices unchanged.

11. What is the standard form of the $2 \times 2$ identity matrix $I_2$?

    $\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Diagonal entries are 1, off-diagonal entries are 0.

12. What is $I_3A$ if $A$ is any $3 \times 3$ matrix?

    $I_3A = A$. Identity matrix preserves any compatible matrix.

13. If $\det(A)=0$ for a square matrix $A$, what can you conclude about $A^{-1}$?

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

14. What determinant condition is equivalent to a square matrix $A$ being invertible?

    $\det(A) \ne 0$. Matrix is invertible if and only if determinant is nonzero.

15. What is the $m \times n$ zero matrix, and what entries does it contain?

    The $m \times n$ matrix with every entry equal to $0$. All entries are zero, regardless of matrix dimensions.

16. What is the standard form of the $3 \times 3$ identity matrix $I_3$?

    $\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$. Diagonal entries are 1, all other entries are 0.

17. Identify the matrix $X$ if $A + X = A$ for all $m \times n$ matrices $A$.

    $X = 0$ (the $m \times n$ zero matrix). Only zero matrix preserves all matrices under addition.

18. What is the additive identity property for matrices using the zero matrix $0$?

    $A + 0 = A$ and $0 + A = A$. Zero matrix acts like 0 in real numbers for addition.

19. What size must the identity matrix be so that $IA = A$ for an $m \times n$ matrix $A$?

    Use $I_m$ on the left: $I_m A = A$. Left multiplication requires $m \times m$ identity for $m \times n$ matrix.

20. What does it mean for a square matrix $A$ to be invertible in terms of an inverse matrix?

    There exists $A^{-1}$ with $AA^{-1}=A^{-1}A=I$. Inverse satisfies this equation, like reciprocals in real numbers.

21. What is the value of $0_{m \times n} + A$ for any $m \times n$ matrix $A$?

    $A$. Addition is commutative; zero matrix leaves $A$ unchanged.

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

    $1$. $3(1)-1(2)=3-2=1$; matrix is invertible.

23. What is the multiplicative identity property for square matrices using $I_n$?

    If $A$ is $n \times n$, then $AI_n = A$ and $I_nA = A$. Identity matrix preserves any square matrix under multiplication.

24. What is the result of multiplying any matrix $A$ by a compatible zero matrix $0$?

    $A^0 = 0$ and $0A = 0$ (when the products are defined). Any matrix times zero matrix yields zero matrix.

25. What size must the identity matrix be so that $AI = A$ for an $m \times n$ matrix $A$?

    Use $I_n$ on the right: $A I_n = A$. Right multiplication requires $n \times n$ identity for $m \times n$ matrix.

26. What is the value of $I_mA$ when $A$ is an $m \times n$ matrix?

    $A$. Left multiplication by identity preserves the matrix.

27. What conclusion can you make if $\det(A) = 0$ for a square matrix $A$?

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

28. What conclusion can you make if $\det(A) = 5$ for a square matrix $A$?

    $A$ is invertible (since $\det(A) \ne 0$). Nonzero determinant (5) guarantees invertibility.

29. What equation defines the additive identity property for matrices $A$ and $0$?

    $A + 0 = A$. Adding zero matrix leaves any matrix unchanged.

30. What is $AI_2$ if $A = \begin{pmatrix} 7 & 0 \\ -2 & 5 \end{pmatrix}$?

    $\begin{pmatrix} 7 & 0 \\ -2 & 5 \end{pmatrix}$. Multiplying by $I_2$ leaves $2 \times 2$ matrix unchanged.

31. What is $A + 0$ if $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$?

    $\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$. Adding zero matrix leaves original unchanged.

32. Identify the matrix $X$ if $AX = A$ for all $n \times n$ matrices $A$.

    $X = I_n$. Only identity matrix preserves all matrices under right multiplication.

33. What is the value of $A + 0_{m \times n}$ for any $m \times n$ matrix $A$?

    $A$. Zero matrix is the additive identity.

34. What determinant condition is equivalent to an $n \times n$ matrix $A$ being invertible?

    $A$ is invertible iff $\det(A) \ne 0$. Nonzero determinant guarantees matrix has inverse.

35. What is the multiplicative identity matrix for $n \times n$ matrices?

    The identity matrix $I_n$. Has 1s on diagonal, 0s elsewhere; acts like 1 in multiplication.

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

    $A$ is invertible iff there exists $A^{-1}$ with $AA^{-1}=A^{-1}A=I_n$. Matrix has inverse that yields identity when multiplied.

37. What is the inverse of the identity matrix $I_n$?

    $I_n^{-1} = I_n$. Identity matrix is its own inverse: $I_n \cdot I_n = I_n$.

38. What is the determinant formula for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$?

    $\det(A)=ad-bc$. Standard formula for $2 \times 2$ determinant.

39. What is the value of $A0_{n \times p}$ when $A$ is $m \times n$ and the product is defined?

    $0_{m \times p}$. Any matrix times zero matrix gives zero matrix.

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