2.3 Inverses of Matrices

• What is the condition for a matrix $A$ to have an inverse $A^{ - 1}$?


  $A \times A^{ - 1} =$$I$
• A matrix $A$ is invertible only if it is non-singular
• Invertible matrices have a determinant equal to zero.

  False
• What must the determinant of a matrix be for it to have an inverse?

  $≠ 0$
• Singular matrices have a determinant equal to zero
• The matrix $A =$$\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$ has an inverse because its determinant is 1.
• Match the type of matrix with its property:

  Non-singular matrix ↔️ Determinant ≠ 0

  Singular matrix ↔️ Determinant = 0
• Which type of matrices are non-invertible?

  Singular matrices
• A matrix with a determinant of 0 is invertible.

  False
• A matrix with a non-zero determinant is called non-singular
• What is the determinant of the matrix $A =$$\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$?


  1
• Under what condition does a matrix $A$ have an inverse $A^{ - 1}$?


  If A is non-singular
• Non-singular matrices have a determinant equal to zero.

  False
• What is the determinant of a non-invertible matrix?

  0
• A matrix is invertible if its determinant is not equal to zero
• What is the formula to calculate the inverse of a 2x2 matrix $A =$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$?


  $A^{ - 1} =$$\frac{1}{|A|} \begin{bmatrix} d & - b \\ - c & a \end{bmatrix}$
• Steps to calculate the inverse of a 2x2 matrix

  1️⃣ Find the determinant|A| = ad - bc</latex>

  2️⃣ Form the adjugate by swapping $a$ and $d$, and negating $b$ and $c$

  3️⃣ Multiply the adjugate by $\frac{1}{|A|}$
• A matrix $A$ has an inverse if $A \times A^{ - 1} =$$A^{ - 1} \times A =$$I$.
• What is the determinant of a2x2 matrix used in the inverse formula?

  $|A| =$$ad - bc$
• How do you calculate the minor $M_{ij}$ in a matrix $A$?


  Determinant of 2x2 submatrix
• The cofactor matrix $C$ applies signs to the minor matrix according to a specific pattern
• The adjoint matrix is obtained by transposing the cofactor matrix.
• What formula is used to calculate the inverse of a matrix $A$ using the adjoint method?


  $A^{ - 1} =$$\frac{1}{|A|} adj(A)$
• Steps to calculate the inverse of a 3x3 matrix using the adjoint method

  1️⃣ Find the determinant

  2️⃣ Find the minor matrix

  3️⃣ Form the cofactor matrix

  4️⃣ Find the adjoint matrix

  5️⃣ Calculate the inverse
• If the determinant of a matrix is 0, the matrix has no inverse.
• The cofactor matrix $C$ applies alternating signs to the minor matrix in a specific pattern
• What is the final step in calculating the inverse of a matrix using the adjoint method?

  Multiply adjoint by $\frac{1}{|A|}$
• Steps for calculating the inverse of a 3x3 matrix using the adjoint method

  1️⃣ Find the determinant of the matrix $|A|$

  2️⃣ Calculate the minor matrix $M$

  3️⃣ Create the cofactor matrix $C$

  4️⃣ Find the adjoint matrix $adj(A)$ by transposing $C$

  5️⃣ Multiply the adjoint matrix by $\frac{1}{|A|}$ to get the inverse $A^{ - 1}$
• The inverse of a matrix $A$ is calculated using the formula $A^{ - 1} =$$\frac{1}{|A|} adj(A)$, where |A|</latex> is the determinant
• If the determinant of a matrix $A$ is zero, the matrix has no inverse.
• The minor $M_{ij}$ of an element $a_{ij}$ in a matrix $A$ is the determinant of the 2x2 submatrix obtained by removing the $i$-th row and $j$-th column
• The cofactor matrix is created by applying alternating signs to the minor matrix.
• The adjoint matrix $adj(A)$ is the transpose of the cofactor matrix.
• Match the type of matrix with its property:

  Non-singular matrix ↔️ Determinant $≠ 0$

  Singular matrix ↔️ Determinant $=$$0$

  Invertible matrix ↔️ Has a non-zero determinant

  Non-invertible matrix ↔️ Has a zero determinant
• The inverse of a matrix $A$ is denoted as $A^{ - 1}$, and it satisfies the condition A \times A^{ - 1} = A^{ - 1} \times A = I</latex>.
• A matrix is invertible if and only if it is non-singular.
• What is the determinant of the matrix $A =$$\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$?


  1
• A matrix with a non-zero determinant has an inverse.
• What condition must a matrix satisfy to be invertible?

  Non-singular
• Match the type of matrix with its determinant condition:

  Invertible matrix ↔️ Determinant ≠ 0

  Non-invertible matrix ↔️ Determinant = 0