Inverse and Determinant of a Matrix - AP Precalculus
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What is the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?
What is the determinant of $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?
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- This is the $2 \times 2$ identity matrix.
- This is the $2 \times 2$ identity matrix.
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What happens to the determinant if two rows of a matrix are swapped?
What happens to the determinant if two rows of a matrix are swapped?
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The determinant changes sign. Row swapping introduces a factor of $-1$ to the determinant.
The determinant changes sign. Row swapping introduces a factor of $-1$ to the determinant.
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If a matrix $A$ is invertible, what can be said about $\det(A)$?
If a matrix $A$ is invertible, what can be said about $\det(A)$?
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It is nonzero. Invertible matrices must have nonzero determinant.
It is nonzero. Invertible matrices must have nonzero determinant.
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What is the determinant of $\begin{bmatrix} a & b \\ b & a \end{bmatrix}$?
What is the determinant of $\begin{bmatrix} a & b \\ b & a \end{bmatrix}$?
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$a^2 - b^2$. Cross multiplication gives $a^2 - b^2$.
$a^2 - b^2$. Cross multiplication gives $a^2 - b^2$.
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Calculate the determinant of $\begin{bmatrix} 6 & 5 \\ 1 & 3 \end{bmatrix}$.
Calculate the determinant of $\begin{bmatrix} 6 & 5 \\ 1 & 3 \end{bmatrix}$.
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- $(6)(3) - (5)(1) = 18 - 5 = 13$.
- $(6)(3) - (5)(1) = 18 - 5 = 13$.
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Find the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.
Find the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.
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-1. This is a permutation matrix with determinant $-1$.
-1. This is a permutation matrix with determinant $-1$.
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State the determinant of $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.
State the determinant of $\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$.
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- $(2)(2) - (1)(1) = 4 - 1 = 3$.
- $(2)(2) - (1)(1) = 4 - 1 = 3$.
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What is the result of $\det(AB)$ if $A$ and $B$ are $2 \times 2$ matrices?
What is the result of $\det(AB)$ if $A$ and $B$ are $2 \times 2$ matrices?
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$\det(A) \cdot \det(B)$. Determinant of products equals product of determinants.
$\det(A) \cdot \det(B)$. Determinant of products equals product of determinants.
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Find the inverse of $\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$.
Find the inverse of $\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}$.
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$\begin{bmatrix} \frac{1}{3} & 0 \\ 0 & \frac{1}{4} \end{bmatrix}$. Diagonal matrix inverse has reciprocals on the diagonal.
$\begin{bmatrix} \frac{1}{3} & 0 \\ 0 & \frac{1}{4} \end{bmatrix}$. Diagonal matrix inverse has reciprocals on the diagonal.
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What is the inverse of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?
What is the inverse of $\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$?
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$\begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{bmatrix}$. For diagonal matrices, invert each diagonal element.
$\begin{bmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{bmatrix}$. For diagonal matrices, invert each diagonal element.
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What is the determinant of $\begin{bmatrix} 5 & 1 \\ 2 & 3 \end{bmatrix}$?
What is the determinant of $\begin{bmatrix} 5 & 1 \\ 2 & 3 \end{bmatrix}$?
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- $(5)(3) - (1)(2) = 15 - 2 = 13$.
- $(5)(3) - (1)(2) = 15 - 2 = 13$.
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Find the inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
Find the inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
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$\begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$. Determinant is $-2$, so multiply adjugate by $-\frac{1}{2}$.
$\begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$. Determinant is $-2$, so multiply adjugate by $-\frac{1}{2}$.
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State the determinant of $2 \times 2$ identity matrix $I_2$.
State the determinant of $2 \times 2$ identity matrix $I_2$.
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- Identity matrices always have unit determinant.
- Identity matrices always have unit determinant.
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Find the determinant of $\begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}$.
Find the determinant of $\begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}$.
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- $(4)(3) - (2)(1) = 12 - 2 = 10$.
- $(4)(3) - (2)(1) = 12 - 2 = 10$.
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How does the determinant of $A^T$ compare to $A$?
How does the determinant of $A^T$ compare to $A$?
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They are equal. Transpose operation preserves determinant value.
They are equal. Transpose operation preserves determinant value.
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What is the determinant of $\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}$?
What is the determinant of $\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix}$?
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- Product of diagonal elements: $7 \times 3 = 21$.
- Product of diagonal elements: $7 \times 3 = 21$.
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What is the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$?
What is the determinant of $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$?
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-1. $(0)(0) - (1)(1) = 0 - 1 = -1$.
-1. $(0)(0) - (1)(1) = 0 - 1 = -1$.
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Calculate the determinant of $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.
Calculate the determinant of $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$.
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- $(2)(4) - (3)(1) = 8 - 3 = 5$.
- $(2)(4) - (3)(1) = 8 - 3 = 5$.
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What is the determinant of a zero matrix?
What is the determinant of a zero matrix?
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- Zero matrices have all zero entries, so determinant is zero.
- Zero matrices have all zero entries, so determinant is zero.
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What is the adjugate of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$?
What is the adjugate of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$?
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$\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$. Swap diagonal elements and negate off-diagonal elements.
$\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$. Swap diagonal elements and negate off-diagonal elements.
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Find the determinant of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
Find the determinant of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
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-2. Using formula: $(1)(4) - (2)(3) = 4 - 6 = -2$.
-2. Using formula: $(1)(4) - (2)(3) = 4 - 6 = -2$.
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What is the result of multiplying a matrix by its inverse?
What is the result of multiplying a matrix by its inverse?
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The identity matrix. Matrix multiplication with its inverse yields the identity matrix.
The identity matrix. Matrix multiplication with its inverse yields the identity matrix.
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Describe what makes a matrix singular.
Describe what makes a matrix singular.
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Its determinant is zero. Singular matrices have zero determinant and no inverse.
Its determinant is zero. Singular matrices have zero determinant and no inverse.
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What is the relationship between the determinants of a matrix and its transpose?
What is the relationship between the determinants of a matrix and its transpose?
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They are equal. Transposition preserves the determinant value.
They are equal. Transposition preserves the determinant value.
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Calculate the determinant of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.
Calculate the determinant of $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix}$.
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- Product of diagonal elements: $3 \times 5 \times 7 = 105$.
- Product of diagonal elements: $3 \times 5 \times 7 = 105$.
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What is the effect on the determinant when a row is multiplied by a scalar $k$?
What is the effect on the determinant when a row is multiplied by a scalar $k$?
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It is multiplied by $k$. Scaling a row scales the entire determinant by that factor.
It is multiplied by $k$. Scaling a row scales the entire determinant by that factor.
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What is the determinant of an identity matrix?
What is the determinant of an identity matrix?
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- Identity matrices always have determinant equal to 1.
- Identity matrices always have determinant equal to 1.
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State the rule for the determinant of a triangular matrix.
State the rule for the determinant of a triangular matrix.
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It is the product of the diagonal elements. Upper or lower triangular matrices have determinant equal to diagonal product.
It is the product of the diagonal elements. Upper or lower triangular matrices have determinant equal to diagonal product.
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What happens to the determinant if two rows of a matrix are swapped?
What happens to the determinant if two rows of a matrix are swapped?
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The determinant changes sign. Row swapping introduces a factor of $-1$ to the determinant.
The determinant changes sign. Row swapping introduces a factor of $-1$ to the determinant.
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Find the inverse of $\begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$.
Find the inverse of $\begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix}$.
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$\frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix}$. Determinant is $24-14=10$, then apply inverse formula.
$\frac{1}{10} \begin{bmatrix} 6 & -7 \\ -2 & 4 \end{bmatrix}$. Determinant is $24-14=10$, then apply inverse formula.
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