State the following determinants for each matrix (just enter the number, with no decimals, in the box). 1. det($I_3$) = 2. The elementary matrix corresponding to the row swap: det$\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$ = 3. The elementary matrix corresponding to the row replacement: det$\begin{pmatrix} 1 & 0 & 0 \\ 7 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ =
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Step 1: The determinant of the identity matrix is always 1. Show more…
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The following theorems are true for square matrices of any size. 1. If every element in a row (or column) of matrix $A$ is $0,$ then $|A|=0$ 2. If the rows of matrix $A$ are the corresponding columns of matrix $B$, then $|B|=|A|$ 3. If any two rows (or columns) of matrix $A$ are interchanged to form matrix $B$, then $|B|=-|A|$ 4. Suppose matrix $B$ is formed by multiplying every element of a row (or column) of matrix $A$ by the real number $k$. Then $|B|=k \cdot|A|$ 5. If two rows (or columns) of a matrix $A$ are identical, then $|A|=0$. 6. Changing a row (or column) of a matrix by adding to it a constant times another row (or column) does not change the determinant of the matrix. Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{lll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 3 & 0 & 0 \end{array}\right|$$
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The following theorems are true for square matrices of any size. 1. If every element in a row (or column) of matrix $A$ is $0,$ then $|A|=0$ 2. If the rows of matrix $A$ are the corresponding columns of matrix $B$, then $|B|=|A|$ 3. If any two rows (or columns) of matrix $A$ are interchanged to form matrix $B$, then $|B|=-|A|$ 4. Suppose matrix $B$ is formed by multiplying every element of a row (or column) of matrix $A$ by the real number $k$. Then $|B|=k \cdot|A|$ 5. If two rows (or columns) of a matrix $A$ are identical, then $|A|=0$. 6. Changing a row (or column) of a matrix by adding to it a constant times another row (or column) does not change the determinant of the matrix. Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrr} 4 & 8 & 0 \\ -1 & -2 & 1 \\ 2 & 4 & 3 \end{array}\right|$$
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