VCAA General Mathematics Matrices

Incorrect. There are more than 2 negative elements; for $i=1$, the elements $f_{12}$, $f_{13}$, and $f_{14}$ are all negative.

Incorrect. This overcounts the negative elements, perhaps by incorrectly evaluating $i=2$ where $i^2 - j = 4 - j \ge 0$ for all valid $j$.

Incorrect. This represents the total number of elements in the $4 \times 4$ matrix ($4 \times 4 = 16$), not just the negative ones.

The Leslie matrix already models only the female population, so this element does not represent the proportion of females.

This element represents a survival rate between age classes, not the proportion of birds that never reproduce.

The reproduction rate for the first year is represented by the element in the first row, first column ($L_{1,1}$), which is 0.

The probability of living the entire lifespan would be the product of the survival rates ($0.2 \times 0.4 = 0.08$), not just the survival rate from year one to year two.

This is incorrect because there are exactly three true statements, not zero.

This is incorrect because there are exactly three true statements, not one.

This is incorrect because there are exactly three true statements, not two.

This is incorrect because the first statement is false. Square matrices with a determinant of zero (singular matrices) do not have an inverse.

Incorrect. Setting $g = -2h$ results in a determinant of $20h$, which is non-zero since $h \neq 0$, meaning the inverse would exist.

Incorrect. Setting $g = h$ results in a determinant of $-4h$, which is non-zero since $h \neq 0$, meaning the inverse would exist.

Incorrect. Setting $g = 2h$ results in a determinant of $-12h$, which is non-zero since $h \neq 0$, meaning the inverse would exist.

Multiplying by 0.15 calculates the amount of the discount itself, not the final discounted price.

Multiplying by 8.5 calculates 850% of the original price, which is a massive price increase rather than a discount.

Multiplying by 15 calculates 1500% of the original price, rather than applying a 15% discount.

Matrix addition requires both matrices to have the exact same dimensions. Since $A$ is a $2 \times 2$ matrix and $B$ is a $3 \times 1$ matrix, the sum $A + B$ is undefined.

The product $BD$ results in a $3 \times 3$ matrix, but $C$ is a $3 \times 2$ matrix. Matrix addition is undefined for matrices of different dimensions.

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Since $D$ has 3 columns and $A$ has 2 rows, the product $DA$ is undefined.

Incorrect. The first row of $P$ has a 1 in the first column, meaning the first letter $t$ remains in its original position.

Incorrect. The fifth row of $P$ has a 1 in the fifth column, meaning the last letter $s$ remains in its original position.

Incorrect. Both $t$ and $s$ remain in their original positions since the first and fifth rows of $P$ have 1s on the main diagonal.

Incorrect. The letter $s$ does not change position because the fifth row of $P$ is $[0, 0, 0, 0, 1]$.

Incorrect. The determinant of $E$ is $mn - (-36) = mn + 36$. For $m=3$ and $n=12$, the determinant is $72$, which is non-zero, meaning the inverse exists.

Incorrect. The determinant of $E$ is $mn + 36$. For $m=12$ and $n=3$, the determinant is $72$, which is non-zero, meaning the inverse exists.

Incorrect. The determinant of $E$ is $mn + 36$. For $m=-3$ and $n=-12$, the determinant is $(-3)(-12) + 36 = 72$, which is non-zero, meaning the inverse exists.

This is incorrect because a permutation matrix must have exactly one 1 in every row and column. Matrix $G$ has two 1s in the second row and zero 1s in the third row.

This is incorrect because an identity matrix requires 1s along the main diagonal and 0s everywhere else. Matrix $G$ has 0s on its main diagonal.

This is incorrect because a diagonal matrix only contains non-zero entries on its main diagonal. Matrix $G$ has non-zero entries off the diagonal, such as the 1 in the first row, second column.

This matrix contains errors in calculating the two-step dominance matrix $D^2$. For example, in row L, it misses Lian's two-step dominances over Maggie and Neil.

This matrix does not represent $T = D + D^2$. It appears to be an incorrectly filled one-step dominance matrix $D$ that does not align with the given win totals.

This is the completed one-step dominance matrix $D$. However, the question asks for the total dominance matrix $T = D + D^2$, which must also include the two-step dominances.

This is a $2 \times 3$ matrix, not a $3 \times 2$ matrix. Additionally, it contains negative values, which is impossible since the elements are defined by a squared difference.

This matrix has the correct elements but incorrect dimensions. It is a $2 \times 3$ matrix (2 rows, 3 columns), whereas the question specifies a $3 \times 2$ matrix.

While the dimensions are correct, the element $k_{12}$ is $-1$. This is incorrect because $(1-2)^2 = (-1)^2 = 1$, and squares cannot be negative.

The element $k_{31}$ is given as $2$, but according to the rule it should be $(3-1)^2 = 2^2 = 4$.

Matrices can only be added if they have the exact same dimensions. The product $JK$ results in a $2 \times 1$ matrix, not $1 \times 3$.

This is the dimension of matrix $J$. Matrix $L$ must match the dimensions of the product $JK$, which is $2 \times 1$.

Matrix $L$ must match the dimensions of the product $JK$. The product of a $2 \times 3$ and a $3 \times 1$ matrix is $2 \times 1$, not $3 \times 2$.

This calculation determines element $c_{11}$ by multiplying the first row of matrix $A$ with the first column of matrix $B$.

This calculation incorrectly pairs the elements of the first row of matrix $A$ with the reversed elements of the first column of matrix $B$.

This calculation incorrectly pairs the elements of the second row of matrix $A$ with the reversed elements of the first column of matrix $B$.

This number corresponds to the number of vertices (teams) in the graph, not the number of edges (games played).

This would be the number of edges if every team played every other team (a complete graph $K_6$), calculated as $\frac{6 \times 5}{2} = 15$.

This is the total number of checkmarks in the table. Because the graph is undirected (Team A vs Team B is the same game as Team B vs Team A), you must divide this sum by 2.