2.2 Determinants

• What is the determinant of a matrix?

  A scalar value
• The determinant indicates whether a matrix is invertible
• How is the determinant of a 2x2 matrix $A =$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ calculated?


  $ad - bc$
• The determinant of \begin{pmatrix} 3 &1 \\ 2 & 4 \end{pmatrix}</latex> is 10.
• What is the formula for calculating the determinant of a 3x3 matrix $B =$$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$?


  $a(ei - fh) - b(di - fg) +$$c(dh - eg)$
• The determinant of \begin{pmatrix} 1 & 2 &3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}</latex> is 0
• Match the matrix type with its determinant formula:

  2x2 Matrix ↔️ $ad - bc$

  3x3 Matrix ↔️ $a(ei - fh) - b(di - fg) +$$c(dh - eg)$
• The determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is calculated as $ad +$$bc$.


  False
• What is the determinant of the matrix $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$?


  $10$
• The determinant of a 3x3 matrix can be calculated using cofactor expansion
• What is the general formula for the cofactor of an element $a_{ij}$?


  $C_{ij} =$$( - 1)^{i + j}M_{ij}$
• The minor of an element is the determinant of the remaining 2x2 matrix.
• What is the determinant of the matrix \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}</latex> when expanded along the first row?

  $0$
• The formula for the determinant of a 3x3 matrix using cofactor expansion is $aC_{11} - bC_{12} +$$cC_{13}$.


  False
• What does the determinant of a matrix indicate about its properties?

  Invertibility and unique solutions
• What is the formula for the determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?


  $ad - bc$
• The determinant of a matrix indicates whether the matrix is invertible or if a system of equations has a unique solution.
• Calculate the determinant of the matrix $\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$.


  $10$
• The determinant of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ is 0.
• What is the formula for the determinant of a 2x2 matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix}</latex>?

  $ad - bc$
• Steps to calculate the determinant of a 3x3 matrix using cofactor expansion

  1️⃣ Choose a row or column

  2️⃣ Calculate the cofactor for each element in the chosen row or column

  3️⃣ Multiply each element by its cofactor

  4️⃣ Sum the results
• How is the cofactor $C_{ij}$ of an element $a_{ij}$ in a 3x3 matrix calculated?


  $C_{ij} =$$( - 1)^{i + j}M_{ij}$
• The minor M_{ij}</latex> of an element $a_{ij}$ is the determinant of the remaining matrix after removing the row and column of $a_{ij}$.
• What is the term used for the 2x2 determinant obtained after removing a row and column from a 3x3 matrix to calculate the determinant?

  $Minor$
• What is the general formula for calculating the determinant of a 3x3 matrix using cofactor expansion along the first row?

  $|A| =$$aC_{11} +$$bC_{12} +$$cC_{13}$
• The cofactor of an element $a_{ij}$ is calculated as $C_{ij} =$$( - 1)^{i + j}M_{ij}$, where $M_{ij}$ is the minor
• The minor $M_{ij}$ is obtained by removing the row and column of $a_{ij}$ and calculating the determinant of the remaining 2x2 matrix.
• Steps to calculate the determinant of a 3x3 matrix using cofactor expansion

  1️⃣ Choose a row or column to expand along

  2️⃣ Calculate the minor of each element in the chosen row or column

  3️⃣ Calculate the cofactor of each element using $C_{ij} =$$( - 1)^{i + j}M_{ij}$

  4️⃣ Multiply each element by its cofactor

  5️⃣ Sum the results to find the determinant
• What is the general formula for calculating the determinant of a3x3 matrix using cofactor expansion along the first row?

  $|A| =$$aC_{11} +$$bC_{12} +$$cC_{13}$
• The cofactor of an element $a_{ij}$ is calculated as C_{ij} = ( - 1)^{i + j}M_{ij}</latex>, where $M_{ij}$ is the minor
• The minor $M_{ij}$ is obtained by removing the row and column of $a_{ij}$ and calculating the determinant of the remaining 2x2 matrix.
• Steps to calculate the determinant of a 3x3 matrix using cofactor expansion

  1️⃣ Choose a row or column to expand along

  2️⃣ Calculate the minor of each element in the chosen row or column

  3️⃣ Calculate the cofactor of each element using $C_{ij} =$$( - 1)^{i + j}M_{ij}$

  4️⃣ Multiply each element by its cofactor

  5️⃣ Sum the results to find the determinant
• What is the general formula for calculating the determinant of a 3x3 matrix using cofactor expansion along the first row?

  $|A| =$$aC_{11} +$$bC_{12} +$$cC_{13}$
• The cofactor of an element $a_{ij}$ is calculated as C_{ij} = ( - 1)^{i + j}M_{ij}</latex>, where $M_{ij}$ is the minor
• The minor $M_{ij}$ is obtained by removing the row and column of $a_{ij}$ and calculating the determinant of the remaining 2x2 matrix.
• Steps to calculate the determinant of a 3x3 matrix using cofactor expansion

  1️⃣ Choose a row or column to expand along

  2️⃣ Calculate the minor of each element in the chosen row or column

  3️⃣ Calculate the cofactor of each element using $C_{ij} =$$( - 1)^{i + j}M_{ij}$

  4️⃣ Multiply each element by its cofactor

  5️⃣ Sum the results to find the determinant
• What is the general formula for calculating the determinant of a 3x3 matrix using cofactor expansion along the first row?

  $|A| =$$aC_{11} +$$bC_{12} +$$cC_{13}$
• The cofactor of an element $a_{ij}$ is calculated as C_{ij} = ( - 1)^{i + j}M_{ij}</latex>, where $M_{ij}$ is the minor
• The minor $M_{ij}$ is obtained by removing the row and column of $a_{ij}$ and calculating the determinant of the remaining 2x2 matrix.
• Steps to calculate the determinant of a 3x3 matrix using cofactor expansion

  1️⃣ Choose a row or column to expand along

  2️⃣ Calculate the minor of each element in the chosen row or column

  3️⃣ Calculate the cofactor of each element using $C_{ij} =$$( - 1)^{i + j}M_{ij}$

  4️⃣ Multiply each element by its cofactor

  5️⃣ Sum the results to find the determinant